Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #11757163⋊C2 (order 1920) · Inn(G) = C24 · Out(G) = S5 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| extraspecial | 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C24 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(4,12)(7,9)(10,16);(1,2,5,6)(3,9,11,7)(4,8,12,13)(10,15,16,14);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)Polycyclic presentation:
2^5|b^2=e,c^2=e,b^a=be,d^a=de,c^b=ce17 conjugacy classes — view the full character table.
| derived subgroup of | 2−1+4⋊C5, C2×2−1+4⋊C5, #585⋊C5, … |
| Fitting subgroup of | 2−1+4⋊C3, 2−1+4⋊C5, #5430, … |
| maximal subgroup of | #454, #456, #577, … |
| maximal normal subgroup of | #454, #456, #577, … |
| maximal quotient of | #519, #520, #536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 50);