Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(17,18,21,27,22,26,31,25)(19,28,30,24,20,23,29,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,26,28)(21,30,31,29)(24,27,32,25);(10,13)(14,16)(17,20,22,19)(18,24,26,32)(21,30,31,29)(23,27,28,25);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(17,21,22,31)(18,25,26,27)(19,29,20,30)(23,32,28,24)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6450092);