Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #36020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1731 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)(11,15)(12,14)(13,16)(17,18,22,27)(19,29,32,24)(20,25,34,40)(21,26,23,28)(30,43,44,37)(31,42,36,38)(33,39,35,41)(45,48,46,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,31)(22,32)(23,36)(25,37)(26,38)(27,29)(28,42)(33,45)(34,44)(35,46)(39,47)(40,43)(41,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,34)(18,25,27,40)(19,30,32,44)(21,35,23,33)(24,37,29,43)(26,41,28,39)(31,46,36,45)(38,48,42,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21)(18,26)(20,33)(22,23)(25,39)(27,28)(30,44)(31,36)(34,35)(37,43)(38,42)(40,41);(2,6)(7,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,33,34,35)(24,38,29,42)(25,39,40,41)(30,45,44,46)(37,47,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fg,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7477007);