Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #72218792 (order 12288) · Inn(G) = C24 · Out(G) = #11714170 |
| 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 | C23 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,10)(8,16)(11,13)(14,20);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)Polycyclic presentation:
2^6|a^2=f,b^2=e,c^2=e,b^a=be,d^a=de,c^b=ce34 conjugacy classes — view the full character table.
| Fitting subgroup of | #519⋊C3, #73074, #85350, … |
| maximal subgroup of | #1830, #1857, #1886, … |
| maximal normal subgroup of | #1830, #1857, #1886, … |
| maximal quotient of | #2274, #2276, #2310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 200);