Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #265227 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24);(1,3)(2,5)(4,7)(6,8)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24);(5,8);(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)Polycyclic presentation:
2^9|b^2=e,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268362, #72268367, #72268370, … |
| maximal normal subgroup of | #72268362, #72268367, #72268370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 172425);