Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C23×#120 (order 256) · Inn(G) = C23 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C23 |
| commutator, socle | C22 | C22×C4 |
| prime radical | C2×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
61 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)(5,6,9,7,8,10,12,11);(2,4)(6,10)(7,11);(5,7,9,10,8,11,12,6)Polycyclic presentation:
2^6|a^2=e,c^2=e,e^2=f,b^a=bd,c^a=cf,c^b=cf28 conjugacy classes — view the full character table.
| maximal subgroup of | #1330, #1581, #1583, … |
| maximal normal subgroup of | #1330, #1581, #1583, … |
| maximal quotient of | #1589, #1590, #1591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 117);