Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | |
918 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)(9,10,13,17)(11,18,21,24)(12,16,14,22)(15,23,19,20);(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,22)(13,17)(14,16)(15,21)(18,23)(20,24);(9,11)(10,15)(12,20)(13,21)(14,23)(16,24)(17,19)(18,22);(2,5)(6,8)(9,12,13,14)(10,16,17,22)(11,20,21,23)(15,24,19,18);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,16,17,22)(11,20,21,23)(15,24,19,18)Polycyclic presentation:
2^8|a^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #221310, #221320, #221331, … |
| maximal normal subgroup of | #221310, #221320, #221331, … |
| maximal quotient of | #6379563, #6379564, #6380395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54481);