Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = #61451 (order 256) · Inn(G) = #102 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #102 |
| commutator | C8 | C2×C4 |
| Frattini, cosocle | C2×C8 | C22 |
| prime radical | C22 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9,6,11,21,10)(3,12,14,25,15,29,20,28)(4,16,18,8,19,26,32,27)(7,17,22,13,24,30,23,31);(1,3,6,15)(2,7,11,24)(4,17,19,30)(5,20,21,14)(8,25,27,28)(9,23,10,22)(12,16,29,26)(13,18,31,32)Polycyclic presentation:
2^6|a^2=d,b^2=f,c^2=e,d^2=f,e^2=f,b^a=bc,c^a=ce,e^a=ef,c^b=cef,d^b=df,e^b=ef16 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#362, #2148, #2180, … |
| maximal normal subgroup of | C2×#362, #2148, #2180, … |
| maximal quotient of | #1334, #1350, #1362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 43);