Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #72256075 (order 12288) · Inn(G) = C23 · Out(G) = C2×#11715850 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C23 |
| commutator | C22 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(6,7)(9,10,11,12);(5,6,8,7);(1,2,3,4)(5,7,8,6)Polycyclic presentation:
2^6|a^2=f,b^2=d,c^2=e,b^a=bd,c^a=ce28 conjugacy classes — view the full character table.
| derived subgroup of | #94670, #94671, #94672, … |
| Frattini subgroup of | #7354, #7355, #7358, … |
| Fitting subgroup of | #384⋊C3, #67999, #72992, … |
| maximal subgroup of | #1296, #1325, #1802, … |
| maximal normal subgroup of | #1296, #1325, #1802, … |
| maximal quotient of | C4⋊#95, #1444, #1842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 65);