Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72210568 (order 2048) · Inn(G) = C24 · Out(G) = #2196 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,12)(9,11)Polycyclic presentation:
2^6|b^a=be,c^a=cf,d^a=de,c^b=ce22 conjugacy classes — view the full character table.
| maximal subgroup of | #1595, #1601, #1607, … |
| maximal normal subgroup of | #1595, #1601, #1607, … |
| maximal quotient of | #2308, #2315, #2321, … |
| outer automorphism group of | C2×#1412, #13594, C2×#1413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 215);