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 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
747 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)(12,16)(13,15)(17,18,22,23)(19,24,25,20)(21,29,30,26)(27,28,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,15,12)(18,23)(20,24)(26,29)(28,31);(2,5)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,3,4,7)(2,6,5,8)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^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 | #7203081, #7203085, #7203145, … |
| maximal normal subgroup of | #7203081, #7203085, #7203145, … |
| maximal quotient of | #582179, #582391, #582404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45968);