Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #72215858 (order 1024) · Inn(G) = C23 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23 |
| commutator, Frattini, socle, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(2,4);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^6|d^2=e,b^a=be,c^a=cf28 conjugacy classes — view the full character table.
| maximal subgroup of | #1501, #1504, #1513, … |
| maximal normal subgroup of | #1501, #1504, #1513, … |
| maximal quotient of | #2280, #2290, #2296, … |
| outer automorphism group of | C2×D16, C2×Q32, C2×Dic10, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 206);