Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,4,8,5)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,31)(21,26)(22,37)(23,27)(25,34)(28,44)(29,39)(30,40)(32,46)(33,43)(35,47)(36,41)(38,45)(42,48);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,32)(21,36)(22,30)(23,29)(25,42)(26,41)(27,39)(28,35)(31,46)(33,45)(34,48)(37,40)(38,43)(44,47);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,20,21,33,23,35,22,34)(18,25,37,47,27,43,26,31)(19,28,36,48,29,32,30,45)(24,38,40,46,39,42,41,44);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(19,29)(24,39)(28,32)(30,36)(38,42)(40,41)(44,46)(45,48);(1,5,6,4)(2,3,7,8)(17,21,23,22)(18,26,27,37)(19,30,29,36)(20,33,35,34)(24,40,39,41)(25,31,43,47)(28,45,32,48)(38,46,42,44)Polycyclic presentation:
2^9|a^2=fhi,c^2=gi,d^2=fhi,e^2=fh,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=dfhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897081);