Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18,23,27)(19,24,30,40)(20,31,33,41)(21,26,22,35)(25,36,42,32)(28,43,45,48)(29,38,34,39)(37,44,47,46);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,34)(22,29)(23,30)(25,37)(26,39)(27,40)(31,43)(32,46)(33,45)(35,38)(36,44)(41,48)(42,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,42)(19,28,30,45)(21,32,22,36)(24,37,40,47)(26,41,35,31)(29,44,34,46)(38,43,39,48);(11,16)(13,15)(17,21,23,22)(18,26,27,35)(19,29,30,34)(20,32,33,36)(24,38,40,39)(25,41,42,31)(28,44,45,46)(37,43,47,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,27)(19,30)(25,42)(26,35)(28,45)(29,34)(31,41)(44,46)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6409406);