Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,18)(11,17,21,15)(12,23,24,34)(14,27,26,19)(16,28,32,25)(20,35,36,40)(22,30,38,33)(29,37,31,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,21)(10,15,18,17)(12,20,24,36)(14,22,26,38)(16,29,32,31)(19,30,27,33)(23,40,34,35)(25,37,28,39);(9,12,13,24)(10,16,18,32)(11,20,21,36)(14,28,26,25)(15,29,17,31)(19,34,27,23)(22,39,38,37)(30,35,33,40);(3,8)(5,7)(9,11)(10,17)(12,20)(13,21)(14,22)(15,18)(16,31)(19,33)(23,35)(24,36)(25,37)(26,38)(27,30)(28,39)(29,32)(34,40);(11,21)(15,17)(20,36)(22,38)(29,31)(30,33)(35,40)(37,39)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,e^b=eh,g^c=gh,e^d=eh55 conjugacy classes — view the full character table.
| maximal subgroup of | #7142811, #7142813, #7142824, … |
| maximal normal subgroup of | #7142811, #7142813, #7142824, … |
| maximal quotient of | #6379091, #6379322, #6379733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54192);