Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #10596770⋊C2 (order 1152) · Inn(G) = C24 · Out(G) = GO+(4,𝔽2) |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| extraspecial | 2+1+4 |
| Heisenberg | H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C24 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 4 generators on 8 points:
(1,2)(3,7)(4,5)(6,8);(2,5)(6,7);(1,2)(3,6)(4,5)(7,8);(1,3)(2,6)(4,8)(5,7)Polycyclic presentation:
2^5|b^a=be,d^a=de,c^b=ce17 conjugacy classes — view the full character table.
| derived subgroup of | #997, B(4,𝔽2)⋊C3, #458⋊C3, … |
| Fitting subgroup of | #994, #997, #5428, … |
| maximal subgroup of | #453, #455, B(4,𝔽2), … |
| maximal normal subgroup of | #453, #455, B(4,𝔽2), … |
| maximal quotient of | #518, #520, #534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 49);