Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 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)(17,18,23,29)(19,31,34,25)(20,27,21,37)(22,30,24,28)(26,38,44,36)(32,41,35,46)(33,42,39,45)(40,48,43,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,35)(21,32)(22,39)(23,34)(24,33)(26,43)(27,41)(28,45)(29,31)(30,42)(36,47)(37,46)(38,48)(40,44);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,32)(21,23)(22,36)(24,38)(25,40)(27,28)(29,44)(30,37)(31,43)(33,47)(34,35)(39,48)(41,42)(45,46);(3,8)(5,7)(11,16)(13,15)(17,21,23,20)(18,27,29,37)(19,32,34,35)(22,36,24,38)(25,41,31,46)(26,30,44,28)(33,48,39,47)(40,45,43,42);(2,6)(5,7)(10,14)(13,15)(17,22,23,24)(18,28,29,30)(19,33,34,39)(20,36,21,38)(25,42,31,45)(26,27,44,37)(32,47,35,48)(40,41,43,46)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7141592);