Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,3)(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(2,3)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(1,3,4,2)(10,14)(13,16)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(5,7,8,6)(18,23)(22,25)(24,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ei,d^2=eh,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266551);