Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | Aut(G) = #72285559 (order 4096) · Inn(G) = #342 · Out(G) = #379 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×#117 | C22 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,25,22,26,35,43)(19,27,29,44,30,39,34,47)(20,24,32,40,33,41,46,48)(23,36,37,28,38,45,42,31);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,23,26,38)(20,31,33,28)(21,34)(22,30)(24,39)(25,42,43,37)(27,41)(29,35)(32,45,46,36)(40,44)(47,48)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=h,e^2=f,b^a=bc,c^a=cf,e^a=egh,c^b=cg,d^b=dfh,g^b=gh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7327, #128168, #128169, … |
| maximal normal subgroup of | C2×#7327, #128168, #128169, … |
| maximal quotient of | #93467, #93497, #93558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 315);