Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
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 | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,16,19,17,20,15,18);(1,2,3,4);(5,7)(6,9)(8,12)(10,11);(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,g^c=gh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264124, #72264127, #72264139, … |
| maximal normal subgroup of | #72264124, #72264127, #72264139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95466);