Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,20)(11,22,25,16)(12,18,27,34)(13,21,15,19)(17,29,33,26)(23,37,39,31)(24,32,28,35)(30,40,36,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,25)(10,16,20,22)(12,23,27,39)(13,28,15,24)(17,30,33,36)(18,31,34,37)(19,35,21,32)(26,40,29,38);(9,12)(10,17)(11,23)(13,26)(14,27)(15,29)(16,30)(18,19)(20,33)(21,34)(22,36)(24,38)(25,39)(28,40)(31,35)(32,37);(3,8)(5,7)(9,12)(10,18)(11,23)(13,29)(14,27)(15,26)(16,31)(17,19)(20,34)(21,33)(22,37)(24,40)(25,39)(28,38)(30,35)(32,36);(2,6)(5,7)(9,13,14,15)(10,19,20,21)(11,24,25,28)(12,26,27,29)(16,32,22,35)(17,18,33,34)(23,38,39,40)(30,37,36,31)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61691, #10577300, #10577331, … |
| maximal normal subgroup of | C2×#61691, #10577300, #10577331, … |
| maximal quotient of | #6386068, #6387736, #6387738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54679);