Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| 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 | |
966 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)(9,10,13,17)(11,19,16,15)(12,21,23,20)(14,24,22,18);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,16)(14,20)(17,19)(21,22)(23,24);(9,12,22,15,13,23,14,19)(10,16,24,20,17,11,18,21);(2,5)(6,8)(11,16)(12,23)(15,19)(20,21);(1,3,4,7)(2,6,5,8)(10,17)(11,16)(18,24)(20,21)Polycyclic presentation:
2^8|a^2=h,c^2=gh,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250457, #250461, #250465, … |
| maximal normal subgroup of | #250457, #250461, #250465, … |
| maximal quotient of | #6386894, #6389518, #6389546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55434);