Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#131 (order 64) · Inn(G) = C22 · Out(G) = C22×C4 |
| holomorph | G ⋊ Aut(G) = #72225136 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
| modular maximal cyclic | M64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C16 | C22 |
| commutator, socle | C2 | C2×C16 |
| prime radical | C22 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)Polycyclic presentation:
2^6|a^2=c,c^2=d,d^2=e,e^2=f,b^a=bf40 conjugacy classes — view the full character table.
| maximal subgroup of | #1398, #1400, #1401, … |
| maximal normal subgroup of | #1398, #1400, #1401, … |
| maximal quotient of | #1397, C22⋊C32, C4⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 51);