Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13094 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,11,14,12,15,22,20)(9,16,18,17,19,10,13,21);(7,9,17,8,12,19,21,15)(10,20,11,13,16,14,22,18);(4,6)74 conjugacy classes — view the full character table.
| maximal subgroup of | #72266765, #72266766, #72266768, … |
| maximal normal subgroup of | #72266765, #72266766, #72266768, … |
| maximal quotient of | C2×#148577, #72266862, #72266888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401923);