Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,20,29)(12,22,26,39)(13,27,24,17)(16,31,33,25)(19,34,23,38)(21,37,35,30)(28,40,32,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,23)(13,21)(14,20)(16,32)(17,30)(18,29)(19,26)(22,38)(24,35)(25,40)(27,37)(28,33)(31,36)(34,39);(9,12,24,31,14,26,13,25)(10,16,27,39,18,33,17,22)(11,19,35,40,20,23,21,36)(15,28,37,38,29,32,30,34);(3,8)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,30)(18,29)(22,34)(23,26)(24,35)(25,36)(27,37)(31,40)(32,33)(38,39);(2,6)(5,7)(11,20)(15,29)(19,23)(21,35)(28,32)(30,37)(34,38)(36,40)Polycyclic presentation:
2^8|a^2=h,c^2=gh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130295, #130297, #486346, … |
| maximal normal subgroup of | #130295, #130297, #486346, … |
| maximal quotient of | #6386766, #6389173, #6389176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55323);