Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,24)(19,26,27,23)(20,28,29,32)(22,31,30,25);(10,13)(14,15)(17,19,21,27)(18,23,24,26)(20,22,29,30)(25,32,31,28);(1,2,3,4)(17,20,22,27,21,29,30,19)(18,23,25,28,24,26,31,32);(9,11,12,16)(10,14,13,15)(18,24)(23,26)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267890);