Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #61451 (order 256) · Inn(G) = C2×D8 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23 |
| prime radical | #135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15)(5,20)(6,8)(7,12)(9,18)(10,17)(13,31)(14,28)(16,26)(19,27)(21,25)(22,30)(23,32)(24,29);(1,3,6,14)(2,7,8,22)(4,16,19,32)(5,21,17,13)(9,25,24,31)(10,26,20,23)(11,27,28,15)(12,29,30,18);(2,8)(7,22)(9,24)(10,20)(11,28)(15,27)(23,26)(25,31)Polycyclic presentation:
2^6|b^2=f,d^2=ef,e^2=f,b^a=bd,c^a=cf,d^a=de,e^a=ef,d^b=de,e^b=ef16 conjugacy classes — view the full character table.
| maximal subgroup of | #2180, #2191, #2194, … |
| maximal normal subgroup of | #2180, #2191, #2194, … |
| maximal quotient of | #2139, #2141, #2159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 191);