Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #72225731 (order 4096) · Inn(G) = C23 · Out(G) = #845297 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#121 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23 |
| commutator, Frattini, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 10 points:
(1,2);(4,6)(8,10);(7,8,9,10);(3,4)(5,6)28 conjugacy classes — view the full character table.
| maximal subgroup of | #1471, C2×#327, #1507, … |
| maximal normal subgroup of | #1471, C2×#327, #1507, … |
| maximal quotient of | C2×#380, C2×#386, C2×#388, … |
| outer automorphism group of | #1302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 203);