Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(9,10)(11,13)(12,15)(14,16)(17,18,21,24,20,22,28,29)(19,25,27,32,26,23,30,31);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(18,22)(23,25)(24,29)(31,32);(1,2,4,3)(6,8)(9,12,14,13)(10,15,16,11)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32);(1,3,4,2)(9,13,14,12)(10,15,16,11)(19,26)(23,25)(27,30)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=dei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292676);