Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = #1202 (order 1024) · Inn(G) = C24 · Out(G) = D42 |
| 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 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(5,6,9,10)(7,8,11,12);(6,10)(8,12);(1,2)(3,4)(5,7,9,11)(6,8,10,12);(5,8)(6,11)(7,10)(9,12)Polycyclic presentation:
2^6|a^2=e,c^2=e,b^a=be,c^a=cf,d^b=de25 conjugacy classes — view the full character table.
| maximal subgroup of | #1625, #1635, #1647, … |
| maximal normal subgroup of | #1625, #1635, #1647, … |
| maximal quotient of | #2339, #2344, D4×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 228);