Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,27,30,24)(20,25,33,36)(21,34,35,26)(28,31,42,37)(29,38,43,40)(32,45,46,39)(41,47,48,44);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,22,30)(18,24,23,27)(20,31,33,37)(21,29,35,43)(25,28,36,42)(26,38,34,40)(32,44,46,47)(39,41,45,48);(10,14)(13,16)(17,20,22,33)(18,25,23,36)(19,28,30,42)(21,32,35,46)(24,37,27,31)(26,39,34,45)(29,41,43,48)(38,47,40,44);(2,5)(6,8)(17,21)(18,26)(19,29)(20,32)(22,35)(23,34)(24,38)(25,39)(27,40)(28,41)(30,43)(31,44)(33,46)(36,45)(37,47)(42,48);(1,3,4,7)(2,6,5,8)(18,23)(19,24)(20,33)(21,35)(27,30)(28,31)(29,40)(37,42)(38,43)(39,45)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6587453);