Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,29,36)(28,35,43,47)(32,39,44,48)(41,45,42,46);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,23,26,37)(20,28,31,43)(21,27,33,29)(24,35,38,47)(25,34,40,36)(32,41,44,42)(39,45,48,46);(9,12,13,11)(10,14,15,16)(19,27)(21,33)(23,34)(25,40)(28,41)(29,30)(32,44)(35,45)(36,37)(39,48)(42,43)(46,47);(2,6)(5,7)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,39)(26,38)(27,41)(29,42)(30,43)(33,44)(34,45)(36,46)(37,47)(40,48);(3,8)(5,7)(19,27)(20,31)(21,33)(23,34)(24,38)(25,40)(28,42)(29,30)(35,46)(36,37)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597367);