Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32);(2,6)(4,7)(9,11,13,10)(12,15,16,14)(18,23)(21,24)(25,28)(29,31);(1,2)(3,7)(4,8)(5,6)(9,12,13,16)(10,14,11,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(1,3,5,8)(2,7,6,4)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32);(1,4,5,7)(2,8,6,3)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=h,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 | #6693323, #6693330, #6693347, … |
| maximal normal subgroup of | #6693323, #6693330, #6693347, … |
| maximal quotient of | #521975, #522501, #523036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33012);