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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 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)(9,10,12,14)(11,13,15,16)(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)(9,11)(10,13)(12,15)(14,16)(20,33)(25,40)(29,34)(35,36)(38,41)(42,43)(44,48)(46,47);(11,15)(13,16)(17,19,21,30,23,32,22,31)(18,24,37,45,27,39,26,28)(20,34,36,46,33,29,35,47)(25,41,42,44,40,38,43,48);(2,6)(5,7)(17,20,23,33)(18,25,27,40)(19,29,32,34)(21,36,22,35)(24,38,39,41)(26,43,37,42)(28,44,45,48)(30,47,31,46);(3,8)(5,7)(11,15)(13,16)(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|a^2=h,c^2=gi,d^2=i,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, 6657276);