Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501370 |
| 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 | |
627 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 627 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:
(1,2,9,38,6,13,8,15)(3,16,36,44,19,10,21,55)(4,11,26,48,24,49,37,56)(5,27,29,80,30,12,39,51)(7,32,40,54,31,14,35,90)(17,57,63,97,62,41,69,93)(18,42,65,95,66,58,85,106)(20,43,87,88,67,59,33,52)(22,45,72,99,71,100,68,96)(23,46,86,107,73,78,75,102)(25,76,77,53,60,47,34,89)(28,50,83,105,84,79,91,123)(61,108,115,125,111,92,112,127)(64,109,81,121,114,94,124,103)(70,98,113,126,116,120,117,128)(74,118,82,122,110,101,119,104);(1,3,6,19)(2,10,13,16)(4,22,24,71)(5,18,30,66)(7,33,31,87)(8,36,9,21)(11,45,49,100)(12,42,27,58)(14,52,32,88)(15,55,38,44)(17,60,62,25)(20,40,67,35)(23,70,73,116)(26,68,37,72)(28,81,84,124)(29,85,39,65)(34,63,77,69)(41,76,57,47)(43,90,59,54)(46,98,78,120)(48,96,56,99)(50,103,79,121)(51,106,80,95)(53,93,89,97)(61,110,111,74)(64,91,114,83)(75,113,86,117)(82,112,119,115)(92,118,108,101)(94,123,109,105)(102,126,107,128)(104,125,122,127);(1,4,8,37,6,24,9,26)(2,11,38,48,13,49,15,56)(3,17,21,69,19,62,36,63)(5,23,29,86,30,73,39,75)(7,34,35,60,31,77,40,25)(10,41,55,93,16,57,44,97)(12,46,80,102,27,78,51,107)(14,53,90,47,32,89,54,76)(18,61,65,115,66,111,85,112)(20,68,33,71,67,72,87,22)(28,82,83,110,84,119,91,74)(42,92,106,125,58,108,95,127)(43,96,88,45,59,99,52,100)(50,104,123,101,79,122,105,118)(64,113,81,116,114,117,124,70)(94,126,121,98,109,128,103,120);(1,5)(2,12)(3,18)(4,23)(6,30)(7,28)(8,29)(9,39)(10,42)(11,46)(13,27)(14,50)(15,51)(16,58)(17,61)(19,66)(20,64)(21,65)(22,70)(24,73)(25,74)(26,75)(31,84)(32,79)(33,81)(34,82)(35,83)(36,85)(37,86)(38,80)(40,91)(41,92)(43,94)(44,95)(45,98)(47,101)(48,102)(49,78)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(59,109)(60,110)(62,111)(63,112)(67,114)(68,113)(69,115)(71,116)(72,117)(76,118)(77,119)(87,124)(88,121)(89,122)(90,123)(93,125)(96,126)(97,127)(99,128)(100,120)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=eh,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,f^c=fi,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, 408568);