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 | #36009 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,39)(20,25,34,42)(21,26,22,27)(29,37,46,48)(30,38,35,43)(32,40,45,47)(33,41,36,44);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,23,31)(18,24,28,39)(20,32,34,45)(21,30,22,35)(25,40,42,47)(26,38,27,43)(29,36,46,33)(37,44,48,41);(10,14)(13,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,41)(27,44)(28,42)(30,45)(31,46)(32,35)(38,47)(39,48)(40,43);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,39,43)(25,41,42,44)(29,45,46,32)(37,47,48,40);(2,5)(6,8)(19,30)(21,22)(24,38)(26,27)(29,45)(31,35)(32,46)(33,36)(37,47)(39,43)(40,48)(41,44)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,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, 6586266);