Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214×C4 |
| direct product | C2×D4×#127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
23274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 14 points:
(1,2);(3,4,5,6);(4,6);(8,10)(12,13);(11,12,14,13);(7,8,9,10)(11,13,14,12)140 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×D4×#127⋊C3 |
| maximal subgroup of | #72213999, #72214454, #1272, … |
| maximal normal subgroup of | #72213999, #72214454, #1272, … |
| maximal quotient of | #1272, C2×D4×#392, D4×#2433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536940);