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 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,15)(11,12,16,14)(17,18,21,22)(19,23,25,29)(20,24,27,30)(26,31,28,32);(2,5)(6,7)(9,11,13,16)(10,14,15,12)(18,22)(19,25)(20,27)(31,32);(10,15)(12,14)(17,19)(18,23)(20,28)(21,25)(22,29)(24,32)(26,27)(30,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,14)(10,11,15,16)(17,20)(18,24)(19,26)(21,27)(22,30)(23,31)(25,28)(29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=g,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×#54203, #8103664, #8103890, … |
| maximal normal subgroup of | C2×#54203, #8103664, #8103890, … |
| maximal quotient of | #586156, #586164, #586500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47191);