Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(20,31)(25,42)(28,33)(32,44)(34,36)(38,43)(45,48)(46,47);(2,6)(7,8)(11,16)(13,15)(17,18,23,27)(19,24,30,41)(20,32,31,44)(21,26,22,37)(25,36,42,34)(28,45,33,48)(29,39,35,40)(38,46,43,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,33)(21,35)(22,29)(23,30)(25,43)(26,40)(27,41)(28,31)(32,48)(34,46)(36,47)(37,39)(38,42)(44,45);(10,14)(13,15)(17,20,21,36,23,31,22,34)(18,25,37,44,27,42,26,32)(19,28,35,46,30,33,29,47)(24,38,39,48,41,43,40,45);(1,4,5,3)(2,7,6,8)(17,21,23,22)(18,26,27,37)(19,29,30,35)(20,34,31,36)(24,39,41,40)(25,44,42,32)(28,46,33,47)(38,45,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fg,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522805);