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 | #36078 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,22)(19,24,30,29)(20,25,33,36)(21,26,34,35)(27,37,41,31)(28,38,42,43)(32,39,45,46)(40,47,48,44);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,27)(26,38)(32,44)(33,37)(34,42)(35,43)(36,41)(39,40)(45,47)(46,48);(10,14)(13,16)(17,20,23,33)(18,25,22,36)(19,27,30,41)(21,32,34,45)(24,37,29,31)(26,39,35,46)(28,40,42,48)(38,47,43,44);(2,6)(5,7)(17,21)(18,26)(19,28)(20,32)(22,35)(23,34)(24,38)(25,39)(27,40)(29,43)(30,42)(31,44)(33,45)(36,46)(37,47)(41,48);(3,8)(5,7)(18,22)(19,29)(21,34)(24,30)(25,36)(27,31)(28,38)(32,45)(37,41)(40,47)(42,43)(44,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^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,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, 6585204);