Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
742 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,26,29)(20,24,28,30)(25,31,27,32);(9,11,12,16)(10,13,14,15)(18,22)(23,29)(24,30)(31,32);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31);(10,14)(13,15)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97006, #131204, #131210, … |
| maximal normal subgroup of | #97006, #131204, #131210, … |
| maximal quotient of | #544316, #544322, #544324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29686);