Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | |
1792 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)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,30,38)(20,24,32,40)(21,25,34,42)(27,35,29,37)(28,36,31,39)(33,41,45,48)(43,46,44,47);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,23,26,38)(20,31,32,28)(21,27,34,29)(24,39,40,36)(25,35,42,37)(33,44,45,43)(41,47,48,46);(9,12,13,11)(10,14,15,16)(19,27)(21,34)(23,35)(25,42)(28,43)(29,30)(31,44)(33,45)(36,46)(37,38)(39,47)(41,48);(2,6)(5,7)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,41)(26,40)(27,43)(29,44)(30,31)(34,45)(35,46)(37,47)(38,39)(42,48);(3,8)(5,7)(19,27)(20,32)(21,34)(23,35)(24,40)(25,42)(28,44)(29,30)(31,43)(36,47)(37,38)(39,46)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597695);