Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#2196 (order 256) · Inn(G) = #120 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 3 generators on 8 points:
(2,5)(3,7)(6,8);(1,2)(3,8)(4,6)(5,7);(3,7)(5,8)Polycyclic presentation:
2^6|d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=ef,d^c=df16 conjugacy classes — view the full character table.
| maximal subgroup of | #1402, #2196, #2198, … |
| maximal normal subgroup of | #1402, #2196, #2198, … |
| maximal quotient of | #1595, #1596, #1600, … |
| holomorph of | D4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 134);