Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 12288 |
| 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 | C23 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,26,22,23,29,30,20,25,28,32,19,24,27,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,22)(18,23,25,24)(21,27,28,29)(26,30,32,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,20,22)(18,24,25,23)(21,27,28,29)(26,31,32,30);(2,8)(6,9)(7,16)(14,15)(18,25)(23,24)(26,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=e,f^2=g,g^2=h,b^a=be,d^a=de,c^b=ceh100 conjugacy classes — view the full character table.
| Fitting subgroup of | #33498⋊C3 |
| maximal subgroup of | #137836, #137838, #137856, … |
| maximal normal subgroup of | #137836, #137838, #137856, … |
| maximal quotient of | #493284, #493324, #493325, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26486);