Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,8)(7,13,14,11)(9,12,17,16)(15,19,20,18)(21,22,26,25)(23,29,31,27)(24,32,34,28)(30,35,33,36);(7,14)(9,17)(11,13)(12,16)(22,25)(27,29)(28,32)(35,36);(1,2,3,4)(5,7,10,14)(6,11,8,13)(9,15,17,20)(12,18,16,19)(21,23)(22,27)(24,33)(25,29)(26,31)(28,36)(30,34)(32,35);(5,8,10,6)(7,11,14,13)(9,16,17,12)(15,18,20,19)(21,24)(22,28)(23,30)(25,32)(26,34)(27,35)(29,36)(31,33);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,22)(23,29,31,27)(24,32,34,28)(30,36,33,35)Polycyclic presentation:
2^8|a^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #482885, #6353245, #6353255, … |
| maximal normal subgroup of | #482885, #6353245, #6353255, … |
| maximal quotient of | #544553, #545098, #545548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35161);