Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | #38032 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1535 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)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,31)(29,44)(30,39)(32,45)(33,40)(34,48)(35,42)(36,43)(38,46)(41,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(20,33)(25,40)(29,34)(35,36)(38,41)(42,43)(44,48)(46,47);(2,6)(3,8)(10,13)(14,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,45)(20,34,35,47,33,29,36,46)(25,41,43,48,40,38,42,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,36)(22,35)(23,33)(24,38)(26,43)(27,40)(28,44)(30,47)(31,46)(32,34)(37,42)(39,41)(45,48);(10,13)(14,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,33,36)(24,28,39,45)(25,42,40,43)(29,46,34,47)(38,48,41,44)Polycyclic presentation:
2^9|c^2=fg,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693601);