Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | C2×#15123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
13222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,6)(12,14)(15,16)(17,18);(8,10)(12,14)(16,18);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18);(3,4,5,6)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#392, C2×#4306523, C2×#772785, … |
| maximal normal subgroup of | C2×D4×#392, C2×#4306523, C2×#772785, … |
| maximal quotient of | #72251076, #72251122, #72255634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515550);