Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#14815 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7802 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)(8,9)(12,13);(7,8,10,9)(16,18);(11,12,14,13)(15,16)(17,18);(3,4)(5,6)(7,9,10,8)(11,13,14,12)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#649261, C2×#773846, C2×#822347, … |
| maximal normal subgroup of | C2×#649261, C2×#773846, C2×#822347, … |
| maximal quotient of | C2×#158362, C2×#154647, C22×#14815 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 514953);