Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
781 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,8,7)(9,10,13,12)(11,14,16,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32);(1,2,4,6)(3,7,8,5)(9,11)(10,14)(12,15)(13,16)(18,23)(21,24)(25,28)(29,31);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,3,4,8)(2,5,6,7)(9,12,13,10)(11,14,16,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fgh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98419, #258620, #258622, … |
| maximal normal subgroup of | #98419, #258620, #258622, … |
| maximal quotient of | #521436, #522202, #522607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32494);