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 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,12,17)(11,16,19,14)(13,15,21,18)(20,23,24,22);(1,3)(2,5)(4,8)(6,7)(9,11,12,19)(10,14,17,16)(13,20,21,24)(15,22,18,23);(10,15)(13,21)(14,22)(16,23)(17,18)(20,24);(3,8)(5,7)(9,11)(10,16)(12,19)(13,20)(14,17)(15,23)(18,22)(21,24);(11,19)(14,16)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=f,b^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 | #7142812, #7142814, #7142823, … |
| maximal normal subgroup of | #7142812, #7142814, #7142823, … |
| maximal quotient of | #6379089, #6379322, #6379731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54189);