Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| 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 | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)(17,18,23,20,22,25,21,30)(19,26,28,32,29,24,31,27);(5,6)(7,8)(10,13)(14,15)(17,19)(18,24)(20,27)(21,31)(22,29)(23,28)(25,26)(30,32);(2,4)(6,8)(17,20,23,25,22,30,21,18)(19,27,28,26,29,32,31,24);(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,20,25,30)(19,28,29,31)(24,32,26,27)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72264164, #72264172, #72264277, … |
| maximal normal subgroup of | #72264164, #72264172, #72264277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280998);