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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1345 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)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(19,28)(20,32)(24,29)(25,42)(31,36)(33,44)(35,37)(40,41);(2,6)(7,8)(11,16)(13,15)(17,18,22,27)(19,29,28,24)(20,33,32,44)(21,26,23,38)(25,37,42,35)(30,45,34,48)(31,41,36,40)(39,47,43,46);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,24,27,29)(20,34,32,30)(21,36,23,31)(25,43,42,39)(26,41,38,40)(33,45,44,48)(35,46,37,47);(10,14)(13,15)(17,20,21,37,22,32,23,35)(18,25,38,44,27,42,26,33)(19,30,36,46,28,34,31,47)(24,39,40,45,29,43,41,48);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,26,27,38)(19,31,28,36)(20,35,32,37)(24,40,29,41)(25,44,42,33)(30,46,34,47)(39,48,43,45)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fg,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,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, 7522782);