Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 196608 |
| 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 | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21);(1,2)(3,6)(4,5)(7,8)(10,12)(14,15)(17,18,20,22)(19,21,24,23);(2,5)(6,8)(9,11,13,16)(10,14,12,15)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,15,16,14)(17,19,20,24)(18,21,22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53004⋊C3 |
| maximal subgroup of | #7221073, #7221077, #7221137, … |
| maximal normal subgroup of | #7221073, #7221077, #7221137, … |
| maximal quotient of | #582294, #582495, #582531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45992);