Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22);(1,2)(3,4)(10,14)(12,15)(18,22)(19,24);(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23);(5,6,7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232088, #72232097, #72232115, … |
| maximal normal subgroup of | #72232088, #72232097, #72232115, … |
| maximal quotient of | #72284694, #72286161, #72286431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 782497);