Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = C2×#120×#525 (order 4096) · Inn(G) = C24 · Out(G) = C2×D4×#41 |
| 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 | |
263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8)(10,14)(15,20)(16,21)(22,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=f,d^2=e,f^2=g,g^2=h,b^a=be,c^a=ch,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #135865, #135871, #136744, … |
| maximal normal subgroup of | #135865, #135871, #136744, … |
| maximal quotient of | #493293, #493295, #493311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26491);