Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
5638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14,16,19,17,20,15,18);(5,7)(6,9)(8,12)(10,11);(2,4);(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,g^b=gh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72237674, #72237676, #72260928, … |
| maximal normal subgroup of | #72237674, #72237676, #72260928, … |
| maximal quotient of | #72261213, #72293047, #72307676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91939);