Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 65536 |
| 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)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(9,11,12,16)(10,13,14,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(20,28)(23,30)(26,32)(29,31);(10,14)(13,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=f,b^2=fh,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 | #131211, #484887, #485462, … |
| maximal normal subgroup of | #131211, #484887, #485462, … |
| maximal quotient of | #544317, #544321, #544322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29685);