Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#10322 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3178 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);(3,4,6,9)(5,7,10,8)(11,12)(13,15)(14,16)(17,18);(4,7)(5,10)(8,9)(11,13)(12,15)(14,17)(16,18);(15,18)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231679, C2×#234325, C2×#248456, … |
| maximal normal subgroup of | #72231679, C2×#234325, C2×#248456, … |
| maximal quotient of | #72261297, #72264345, #72264359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141474);