Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = #72214076 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | C2×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,13)(8,16)(9,11)(10,14)(12,17)(15,19)(18,20);(5,7,8,14,9,15,17,20)(6,10,12,13,11,18,16,19);(1,2,3,4)(6,11)(10,18)(12,16)(13,19)Polycyclic presentation:
2^6|b^2=d,c^2=ef,d^2=f,b^a=bd,c^a=ce,d^a=df22 conjugacy classes — view the full character table.
| maximal subgroup of | #1360, #2210, #2211, … |
| maximal normal subgroup of | #1360, #2210, #2211, … |
| maximal quotient of | #1706, #1707, #1708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 176);