Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,18,21,20)(19,23,24,22);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(18,20)(22,23);(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97169, #130090, #133461, … |
| maximal normal subgroup of | #97169, #130090, #133461, … |
| maximal quotient of | #557185, #557532, #557673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39306);