Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) = #72219852 (order 15360) · Inn(G) = C24 · Out(G) = C3⋊#66297 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,9)(4,12)(6,10)(7,14)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30);(1,2,5,8)(3,10,9,6)(4,7,12,14)(11,16,15,13)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32);(1,3,5,9)(2,6,8,10)(4,11,12,15)(7,13,14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,4)(2,7)(3,11)(5,12)(6,13)(8,14)(9,15)(10,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^6|a^2=e,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de19 conjugacy classes — view the full character table.
| derived subgroup of | #564⋊C3, #564⋊C5, #1658⋊C3, … |
| Fitting subgroup of | #564⋊C3, #564⋊C5, #10617013, … |
| maximal subgroup of | #1658, #1663, C2×#564, … |
| maximal normal subgroup of | #1658, #1663, C2×#564, … |
| maximal quotient of | #2382, #2719, #2726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 245);