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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16);(10,13)(11,15)(17,18,23,26)(19,27,29,30)(20,25,22,31)(21,28,24,32);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,28)(21,26)(22,32)(23,29)(25,30)(27,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,22)(18,25,26,31)(19,28,29,32)(21,27,24,30);(1,4,5,3)(2,8,6,7)(17,21,23,24)(18,19,26,29)(20,30,22,27)(25,32,31,28)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=gi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6638219);