Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C2×#120×#525 (order 4096) · Inn(G) = C24 · Out(G) = C22×D42 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | C2×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,13,17,20,23,14,11,15,18,21,19,22,24);(2,5)(6,8);(9,11,12,18,13,19,20,24,14,10,15,16,21,17,22,23);(1,3,4,7)(2,6,5,8)(10,11)(16,18)(17,19)(23,24)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=e,f^2=g,g^2=h,b^a=be,d^a=dh,d^c=dh100 conjugacy classes — view the full character table.
| maximal subgroup of | #128652, #128655, #135073, … |
| maximal normal subgroup of | #128652, #128655, #135073, … |
| maximal quotient of | #493195, #493196, D4×#346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26472);