Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16);(2,4)(10,13)(14,16)(17,18)(19,22)(20,21)(23,24);(2,4)(18,21)(22,24);(6,8)(17,19,20,23)(18,22,21,24);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264483, #72280075, #72280376, … |
| maximal normal subgroup of | #72264483, #72280075, #72280376, … |
| maximal quotient of | #72285525, #72285960, #72298426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515106);