Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = C2×D4×#120 (order 512) · Inn(G) = C2×D4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator | C4 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4)(6,9)(7,11)(10,12);(5,6)(7,12)(8,10)(9,11);(6,10)(9,12)Polycyclic presentation:
2^6|a^2=e,d^2=f,b^a=bd,d^a=df,c^b=cf,d^b=df22 conjugacy classes — view the full character table.
| maximal subgroup of | #1792, #1793, #1815, … |
| maximal normal subgroup of | #1792, #1793, #1815, … |
| maximal quotient of | #1477, #1483, #1489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 98);