Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17447 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
4608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12,15,18,16,13,14,17);(11,13)(12,16)(14,18)(15,17);(4,7)(8,10);(11,14,16,15)(12,17,13,18)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6343006, C2×#186562, #72239503, … |
| maximal normal subgroup of | C2×#6343006, C2×#186562, #72239503, … |
| maximal quotient of | #72239495, #72309358 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250114);