Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = #6628842 (order 512) · Inn(G) = C2×D4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | C23 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,12)(7,13,15,20)(8,17,14,11)(10,18,19,16);(5,7,14,18,9,15,8,16)(6,10,17,20,12,19,11,13);(2,4)(7,15)(10,19)(13,20)(16,18)Polycyclic presentation:
2^6|a^2=f,b^2=df,d^2=f,b^a=bd,c^a=ce,d^a=df,c^b=cf16 conjugacy classes — view the full character table.
| maximal subgroup of | #1340, #1342, #2191, … |
| maximal normal subgroup of | #1340, #1342, #2191, … |
| maximal quotient of | #1689, #1690, #1695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 151);