Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,35,22,27,21,26)(19,24,36,46,29,38,30,39)(20,31,32,25,34,40,33,41)(28,42,44,37,43,48,45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,36)(25,37)(26,39)(27,38)(31,42)(32,44)(33,45)(34,43)(35,46)(40,48)(41,47);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,32)(22,34)(23,33)(24,37)(26,40)(27,41)(29,43)(30,44)(31,35)(36,45)(38,47)(39,48)(42,46);(2,6)(5,7)(19,29)(24,38)(28,43)(30,36)(37,47)(39,46)(42,48)(44,45);(10,14)(13,15)(17,21,22,23)(18,26,27,35)(19,30,29,36)(20,32,34,33)(24,39,38,46)(25,40,41,31)(28,44,43,45)(37,48,47,42)Polycyclic presentation:
2^9|a^2=gi,c^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6665009);