Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | |
1589 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,31)(21,26)(22,37)(23,27)(25,33)(28,45)(29,39)(30,40)(32,48)(34,43)(35,41)(36,44)(38,47)(42,46);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,32)(21,35)(22,29)(23,30)(25,42)(26,41)(27,40)(28,34)(31,48)(33,46)(36,47)(37,39)(38,44)(43,45);(10,14)(11,16)(17,20,22,33,23,34,21,36)(18,25,26,31,27,44,37,43)(19,28,29,47,30,32,35,46)(24,38,41,45,40,42,39,48);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,29,30,35)(20,33,34,36)(24,39,40,41)(25,43,44,31)(28,46,32,47)(38,45,42,48);(1,3,4,7)(2,6,5,8)(19,30)(24,40)(28,32)(29,35)(38,42)(39,41)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=hi,c^2=ghi,d^2=i,e^2=fhi,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=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6757236);