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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,15,18)(11,16,12,21)(13,19,14,23)(17,20,24,22);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,18,21)(13,22,14,20)(17,23,24,19);(9,12,15,11)(10,17,18,24)(13,22,14,20)(16,23,21,19);(2,5)(6,8)(10,18)(16,21)(17,24)(19,23);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,19,18,23)(11,20,12,22)(16,24,21,17)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=dh,g^a=gh,d^b=df,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61518, #10576809, #10576907, … |
| maximal normal subgroup of | C2×#61518, #10576809, #10576907, … |
| maximal quotient of | #6379582, #6379583, #6380432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54506);