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 | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,26)(21,27)(22,38)(23,29)(24,30)(28,31)(32,41)(33,43)(34,44)(35,45)(36,42)(37,47)(39,48)(40,46);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,30,43)(20,34,36,37)(21,32,23,39)(22,35,31,40)(26,44,42,47)(27,41,29,48)(28,46,38,45);(10,14)(13,15)(17,20,24,36)(18,26,30,42)(19,31,33,22)(21,35,23,40)(25,28,43,38)(27,45,29,46)(32,37,39,34)(41,47,48,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,32,33,39)(20,35,36,40)(22,34,31,37)(25,41,43,48)(26,45,42,46)(28,47,38,44);(2,5)(6,8)(11,16)(13,15)(17,22,24,31)(18,28,30,38)(19,20,33,36)(21,37,23,34)(25,42,43,26)(27,44,29,47)(32,40,39,35)(41,45,48,46)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6606287);