Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C216 |
| direct product | C22×#2408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
41234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(6,7)(14,16);(5,6,8,7)(10,12)(14,16);(9,10)(11,12)(13,14)(15,16);(5,7,8,6)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72231639, C22×#35680, #72231702, … |
| maximal normal subgroup of | #72231639, C22×#35680, #72231702, … |
| maximal quotient of | C2×#6442656, C2×#6442515, #72233556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535031);