Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,15)(11,12,16,14)(17,18)(19,20)(21,22)(23,24);(1,2)(3,7)(4,5)(6,8)(9,11,13,16)(10,14,15,12)(18,22)(19,23);(2,5)(6,7)(10,15)(12,14)(17,19,21,23)(18,20,22,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,23)(19,22)(21,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131578, #484784, #484840, … |
| maximal normal subgroup of | #131578, #484784, #484840, … |
| maximal quotient of | #544555, #545067, #545435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33650);