Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465214 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,23,28,41)(12,29)(13,19,38,50)(14,20)(15,21)(16,22)(17,26,42,27)(18,33)(24,43,54,59)(25,51)(30,61,53,39)(31,44,64,60)(32,46)(34,45)(35,47,63,70)(36,48,37,49)(40,58)(52,66)(55,72)(56,67)(57,68)(62,65)(69,71);(1,3,6,4)(2,7,5,8)(9,11,15,28)(10,17)(12,30)(13,25,38,57)(14,26,16,27)(18,43,29,59)(19,40)(20,23)(21,42)(22,41)(24,32)(31,62)(33,53)(34,54)(35,52)(36,56,37,58)(39,46,61,45)(44,69,60,66)(47,65,70,72)(48,51)(49,68)(50,67)(55,64)(63,71);(1,4,6,3)(2,7,5,8)(9,12)(10,18)(11,24)(13,35)(14,34)(15,33)(16,32)(17,39)(19,47)(20,46)(21,29)(22,45)(23,43)(25,55)(26,30)(27,53)(28,54)(31,36)(37,64)(38,63)(40,66)(41,59)(42,61)(44,48)(49,60)(50,70)(51,65)(52,56)(57,62)(58,71)(67,69)(68,72);(1,5,6,2)(3,8,4,7)(9,13,14,36,15,38,16,37)(10,19,20,48,21,50,22,49)(11,25,26,56,28,57,27,58)(12,31,32,35,33,64,34,63)(17,40,23,51,42,67,41,68)(18,44,45,47,29,60,46,70)(24,52,53,55,54,71,30,62)(39,65,59,66,61,72,43,69)Polycyclic presentation:
2^9|a^2=efgi,b^2=egh,c^2=eg,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,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, 382039);