Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(18,21)(19,22)(23,24);(2,5)(6,8);(10,13)(14,16)(17,18,20,21)(19,22,23,24);(9,11,12,15)(10,14,13,16)(17,19,20,23)(18,22,21,24);(1,3,4,7)(2,6,5,8)(19,23)(22,24)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72283138, #72283191, #72283407 |
| maximal normal subgroup of | #72283138, #72283191, #72283407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537652);