Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | |
916 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)(3,7)(9,10,15,20)(11,16,12,22)(13,19,14,18)(17,21,24,23);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,20,22)(13,23,14,21)(17,18,24,19);(9,12,15,11)(10,17,20,24)(13,23,14,21)(16,18,22,19);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,21,12,23)(16,17,22,24);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,19,20,18)(11,21,12,23)(16,24,22,17)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250332, #250337, #250364, … |
| maximal normal subgroup of | #250332, #250337, #250364, … |
| maximal quotient of | #6386473, #6388494, #6388614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55082);