Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #120×#1020 (order 3072) · Inn(G) = C24 · Out(G) = D4×S4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C22 | C24 |
| commutator | C2 | C23×C4 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11);(5,6)(7,10)(8,9)(11,12);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^6|a^2=f,d^2=e,b^a=be,d^a=de,c^b=ce34 conjugacy classes — view the full character table.
| Fitting subgroup of | #520⋊C3, #73075, #73076, … |
| maximal subgroup of | #1859, #1860, #1882, … |
| maximal normal subgroup of | #1859, #1860, #1882, … |
| maximal quotient of | #2274, #2275, #2276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 201);