Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,12,16)(18,22)(19,25)(24,28)(27,31);(2,5)(6,8)(9,12,14,10)(11,15,13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,6)(4,5)(7,8)(9,13,14,11)(10,15,12,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(1,3)(2,6)(4,7)(5,8)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=fg,d^2=fg,e^2=f,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.
| maximal subgroup of | C2×#52602, #8092280, #8092471, … |
| maximal normal subgroup of | C2×#52602, #8092280, #8092471, … |
| maximal quotient of | #582297, #582500, #582550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45590);