Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #87524 (order 384) · Inn(G) = C23 · Out(G) = C2×S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 3 generators on 8 points:
(1,2)(3,5)(4,6)(7,8);(2,5)(6,8);(2,6)(5,8)Polycyclic presentation:
2^5|b^a=bd,c^a=ce14 conjugacy classes — view the full character table.
| Fitting subgroup of | #120⋊C3, #4625, #1020⋊C2, … |
| maximal subgroup of | AΣL(1,𝔽16), B(4,𝔽2), C2×#120, … |
| maximal normal subgroup of | AΣL(1,𝔽16), B(4,𝔽2), C2×#120, … |
| maximal quotient of | #379, #385, #386, … |
| automorphism group of | C22⋊C4, C4⋊C4 |
| outer automorphism group of | #97, C2×D8, C2×Q16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 27);