Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32930 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #585 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
920 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 920 conjugacy classes of subgroups — too many to draw. Only its 184 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:
(1,2,8,14)(3,15,21,51)(4,10,24,40)(5,11,27,43)(6,29,31,76)(7,32,30,13)(9,33,37,19)(12,45,46,16)(17,47,57,93)(18,48,60,96)(20,63,62,50)(22,64,66,38)(23,68,70,116)(25,71,69,41)(26,72,74,117)(28,75,73,44)(34,77,81,56)(35,78,83,59)(36,52,49,79)(39,87,88,53)(42,89,90,54)(55,101,103,91)(58,106,105,94)(61,108,107,97)(65,112,114,115)(67,86,113,111)(80,102,109,118)(82,98,92,119)(84,99,95,120)(85,100,110,122)(104,124,127,126)(121,128,123,125);(1,3,16,52,7,20,31,33,8,21,45,79,30,62,6,19)(2,9,29,50,13,36,46,51,14,37,76,63,32,49,12,15)(4,17,53,98,25,58,70,77,24,57,87,119,69,105,23,56)(5,18,54,99,28,61,74,78,27,60,89,120,73,107,26,59)(10,34,68,94,41,82,88,93,40,81,116,106,71,92,39,47)(11,35,72,97,44,84,90,96,43,83,117,108,75,95,42,48)(22,55,100,125,67,104,114,118,66,103,122,128,113,127,65,102)(38,80,115,124,86,121,110,101,64,109,112,126,111,123,85,91);(1,4)(2,10)(3,17)(5,22)(6,23)(7,25)(8,24)(9,34)(11,38)(12,39)(13,41)(14,40)(15,47)(16,53)(18,55)(19,56)(20,58)(21,57)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,77)(35,80)(36,82)(37,81)(42,85)(43,64)(44,86)(45,87)(46,88)(48,91)(49,92)(50,94)(51,93)(52,98)(54,100)(59,102)(60,103)(61,104)(62,105)(63,106)(72,115)(73,113)(74,114)(75,111)(76,116)(78,118)(79,119)(83,109)(84,121)(89,122)(90,110)(95,123)(96,101)(97,124)(99,125)(107,127)(108,126)(112,117)(120,128);(1,5)(2,11)(3,18)(4,22)(6,26)(7,28)(8,27)(9,35)(10,38)(12,42)(13,44)(14,43)(15,48)(16,54)(17,55)(19,59)(20,61)(21,60)(23,65)(24,66)(25,67)(29,72)(30,73)(31,74)(32,75)(33,78)(34,80)(36,84)(37,83)(39,85)(40,64)(41,86)(45,89)(46,90)(47,91)(49,95)(50,97)(51,96)(52,99)(53,100)(56,102)(57,103)(58,104)(62,107)(63,108)(68,115)(69,113)(70,114)(71,111)(76,117)(77,118)(79,120)(81,109)(82,121)(87,122)(88,110)(92,123)(93,101)(94,124)(98,125)(105,127)(106,126)(112,116)(119,128)Polycyclic presentation:
2^9|a^2=i,b^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419163);