Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,14,18)(11,15,20,28)(12,22,23,37)(13,25,26,17)(16,30,31,24)(19,32,34,40)(21,36,35,29)(27,38,39,33);(1,2)(3,6)(4,5)(7,8)(9,11,14,20)(10,15,18,28)(12,19,23,34)(13,21,26,35)(16,27,31,39)(17,29,25,36)(22,32,37,40)(24,33,30,38);(9,12,13,24,14,23,26,30)(10,16,17,22,18,31,25,37)(11,19,21,33,20,34,35,38)(15,27,29,32,28,39,36,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(11,20)(12,23)(15,28)(16,31)(21,35)(22,37)(24,30)(29,36)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #244549, #244553, #244558, … |
| maximal normal subgroup of | #244549, #244553, #244558, … |
| maximal quotient of | #6386901, #6389531, #6389554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55448);