Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(2,5)(3,7)(9,10,16,22,15,21,14,20)(11,23,28,17,27,43,26,42)(12,29,39,46,33,18,32,47)(13,34,38,19,37,50,36,49)(24,51,54,66,53,44,30,40)(25,41,56,68,55,67,35,48)(31,59,63,57,62,64,61,45)(52,70,72,69,60,65,58,71);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,30)(13,35)(14,28)(15,27)(16,26)(18,44)(19,48)(20,43)(21,42)(22,23)(24,32)(25,36)(29,51)(31,60)(33,54)(34,41)(37,56)(38,55)(39,53)(40,46)(45,70)(47,66)(49,68)(50,67)(52,62)(57,65)(58,61)(59,71)(63,72)(64,69);(1,3,4,7)(2,6,5,8)(9,12,15,33)(10,18,21,29)(11,24,27,53)(13,31,37,62)(14,32,16,39)(17,40,42,66)(19,45,49,57)(20,46,22,47)(23,44,43,51)(25,52,55,60)(26,30,28,54)(34,64,50,59)(35,58,56,72)(36,61,38,63)(41,65,67,70)(48,69,68,71);(2,5)(6,8)(9,13,14,36,15,37,16,38)(10,19,20,34,21,49,22,50)(11,25,26,35,27,55,28,56)(12,31,32,61,33,62,39,63)(17,41,23,48,42,67,43,68)(18,45,46,64,29,57,47,59)(24,52,30,58,53,60,54,72)(40,65,44,69,66,70,51,71)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339666);