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) = 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 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(10,12)(11,15)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(2,3)(6,7)(9,10,13,12)(11,16,15,14)(19,26)(21,29)(23,25)(24,28);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,16,15,14)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=f,d^2=g,e^2=fh,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 | #6696860, #6696865, #6696925, … |
| maximal normal subgroup of | #6696860, #6696865, #6696925, … |
| maximal quotient of | #544612, #545154, #545633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35839);