Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501366 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24531 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#578 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 628 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2)(3,17,22,10)(4,23,26,58)(5,29,32,12)(6,14)(7,35)(8,16)(9,13)(11,27,49,41)(15,39)(18,62)(19,43)(20,57,40,44)(21,45,73,61)(24,74)(25,75)(28,55,63,76)(30,52,33,50)(31,51,90,86)(34,56)(36,54,93,92)(37,94,84,48)(38,53)(42,66)(46,79)(47,81)(59,67)(60,71)(64,111,113,127)(65,96)(68,97)(69,110)(70,99)(72,102)(77,119,121,128)(78,101)(80,103)(82,107)(83,85)(87,105,91,108)(88,126)(89,125)(95,114,109,115)(98,118)(100,117,124,122)(104,123)(106,116)(112,120);(1,3,9,22)(2,10,13,17)(4,24,26,78)(5,19,32,71)(6,20,8,40)(7,36,39,93)(11,46,49,102)(12,43,29,60)(14,44,16,57)(15,54,35,92)(18,63,65,28)(21,34,73,38)(23,74,58,101)(25,77,80,121)(27,72,41,79)(30,68,33,70)(31,88,90,116)(37,67,84,66)(42,94,59,48)(45,53,61,56)(47,100,85,124)(50,97,52,99)(51,106,86,126)(55,62,76,96)(64,112,113,82)(69,91,118,87)(75,119,103,128)(81,122,83,117)(89,114,123,115)(95,125,109,104)(98,108,110,105)(107,111,120,127);(1,4,8,41,9,26,6,27)(2,11,14,23,13,49,16,58)(3,18,20,66,22,65,40,67)(5,25,30,81,32,80,33,83)(7,37,38,63,39,84,34,28)(10,42,57,62,17,59,44,96)(12,47,52,75,29,85,50,103)(15,55,53,48,35,76,56,94)(19,64,70,115,71,113,68,114)(21,72,36,78,73,79,93,24)(31,89,87,112,90,123,91,82)(43,95,97,111,60,109,99,127)(45,74,92,46,61,101,54,102)(51,107,108,104,86,120,105,125)(69,117,88,121,118,122,116,77)(98,119,126,100,110,128,106,124);(1,5)(2,12)(3,19)(4,25)(6,33)(7,31)(8,30)(9,32)(10,43)(11,47)(13,29)(14,52)(15,51)(16,50)(17,60)(18,64)(20,70)(21,69)(22,71)(23,75)(24,77)(26,80)(27,83)(28,82)(34,91)(35,86)(36,88)(37,89)(38,87)(39,90)(40,68)(41,81)(42,95)(44,99)(45,98)(46,100)(48,104)(49,85)(53,108)(54,106)(55,107)(56,105)(57,97)(58,103)(59,109)(61,110)(62,111)(63,112)(65,113)(66,115)(67,114)(72,117)(73,118)(74,119)(76,120)(78,121)(79,122)(84,123)(92,126)(93,116)(94,125)(96,127)(101,128)(102,124)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,g^b=gh,h^b=hi,e^c=ei,g^c=ghi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408564);