Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #511790 (order 512) · Inn(G) = C22⋊C4 · Out(G) = #120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊C4 |
| commutator | C22 | C42 |
| Frattini, prime radical, cosocle | C24 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 12 points:
(5,6)(7,11,12,9)(8,10);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)Polycyclic presentation:
2^6|a^2=d,b^2=e,b^a=bc,c^a=cf,d^b=df22 conjugacy classes — view the full character table.
| maximal subgroup of | #1343, #1344, #1351, … |
| maximal normal subgroup of | #1343, #1344, #1351, … |
| maximal quotient of | #1280, #1282, #1284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 23);