Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,26)(21,23)(22,25)(27,31)(28,32)(29,30);(2,3)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31);(11,12)(15,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,13,16)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,15,12,16)(17,22,23,30)(18,25,21,29)(19,27,20,28)(24,31,26,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=g,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#54777, #8099791, #8100290, … |
| maximal normal subgroup of | C2×#54777, #8099791, #8100290, … |
| maximal quotient of | #585451, #586714, #587022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47765);