Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
805 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)(11,14)(12,13)(15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(1,2)(3,6)(4,5)(7,8)(10,12)(11,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(2,5)(6,8)(9,11,13,15)(10,14,12,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31);(1,3)(2,6)(4,7)(5,8)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=f,e^2=g,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 | #132878, #486866, #486994, … |
| maximal normal subgroup of | #132878, #486866, #486994, … |
| maximal quotient of | #582251, #582473, #582660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45681);