Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,32,40)(20,25,34,42)(21,26,22,27)(29,37,46,48)(30,38,45,47)(31,39,36,44)(33,41,35,43);(2,5)(3,8)(17,19,23,32)(18,24,28,40)(20,30,34,45)(21,31,22,36)(25,38,42,47)(26,39,27,44)(29,35,46,33)(37,43,48,41);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,41)(27,43)(28,42)(30,36)(31,45)(32,46)(38,44)(39,47)(40,48);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,38)(26,43)(27,41)(28,42)(29,36)(31,46)(32,45)(37,44)(39,48)(40,47);(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,34,35)(24,39,40,44)(25,41,42,43)(29,45,46,30)(37,47,48,38)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414927);