Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | |
1078 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)(9,10,13,19)(11,17,22,15)(12,20,14,18)(16,23,24,21);(1,3)(2,5)(4,8)(6,7)(9,11,13,22)(10,15,19,17)(12,23,14,21)(16,18,24,20);(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,18)(19,24);(3,8)(5,7)(9,11)(10,17)(12,23)(13,22)(14,21)(15,19)(16,18)(20,24);(9,12,13,14)(10,18,19,20)(11,21,22,23)(15,16,17,24)Polycyclic presentation:
2^8|a^2=f,b^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6575780, #6575784, #6575786, … |
| maximal normal subgroup of | #6575780, #6575784, #6575786, … |
| maximal quotient of | #6379103, #6379330, #6379725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54211);