Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,12,15)(18,22)(20,29)(23,25)(26,32);(2,3)(6,7)(11,16)(14,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,14,16,15)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=fgh,e^2=f,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 | #132595, #487608, #488424, … |
| maximal normal subgroup of | #132595, #487608, #488424, … |
| maximal quotient of | #585366, #585756, #587637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51278);