Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#9463 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,7,9,10)(6,8)(11,12,16,18,15,13,14,17);(3,5,6,9)(4,7)(8,10)(11,13,15,12)(14,17,16,18);(14,16)(17,18)74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#133396, C2×#249319, #72241252, … |
| maximal normal subgroup of | C2×#133396, C2×#249319, #72241252, … |
| maximal quotient of | #72241068, C2×#139504, #72284458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108473);