Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24);(2,5)(6,8)(10,14)(12,15)(18,20)(19,23);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,19,21,23)(18,22,20,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,20)(19,22,23,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^8|c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53240⋊C3 |
| maximal subgroup of | C2×#53240, #8105802, #8105898, … |
| maximal normal subgroup of | C2×#53240, #8105802, #8105898, … |
| maximal quotient of | #606750, #606919, #606950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46228);