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 | |
953 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)(9,10,14,19)(11,15,21,24)(12,18,13,17)(16,22,23,20);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,17)(18,23)(19,24);(9,11)(10,16)(12,20)(13,22)(14,21)(15,17)(18,24)(19,23);(2,5)(6,8)(9,12,14,13)(10,17,19,18)(11,20,21,22)(15,23,24,16);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,18,19,17)(11,20,21,22)(15,16,24,23)Polycyclic presentation:
2^8|a^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 | #250327, #250342, #250359, … |
| maximal normal subgroup of | #250327, #250342, #250359, … |
| maximal quotient of | #6386470, #6388493, #6388600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55076);