Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 131072 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(11,15)(14,16)(17,18,22,24)(19,25,27,20)(21,29,30,23)(26,28,32,31);(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,27)(21,23)(22,24)(26,31)(28,32)(29,30);(2,5)(6,7)(10,13)(14,16)(17,19,22,27)(18,20,24,25)(21,26,30,32)(23,28,29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=g,c^2=f,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.
| maximal subgroup of | #132524, #132576, #487315, … |
| maximal normal subgroup of | #132524, #132576, #487315, … |
| maximal quotient of | #582158, #582375, #582575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45956);