Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1083 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)(11,16)(12,24)(13,18)(14,27)(15,19)(17,25)(20,34)(21,30)(22,31)(23,37)(26,32)(28,33)(29,36)(35,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,34)(25,36)(26,35)(27,37)(28,38)(32,40)(33,39);(3,8)(5,7)(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,32)(19,33)(21,35)(22,36)(23,38)(24,27)(30,39)(31,40)(34,37);(2,6)(5,7)(11,21)(16,30)(20,35)(22,23)(29,39)(31,37)(34,40)(36,38);(9,13,15,14)(10,18,19,27)(11,22,21,23)(12,25,26,28)(16,31,30,37)(17,32,33,24)(20,36,35,38)(29,40,39,34)Polycyclic presentation:
2^8|e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131279, #131284, #131321, … |
| maximal normal subgroup of | #131279, #131284, #131321, … |
| maximal quotient of | #6374163, #6374864, #6375507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53737);