Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #2699 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(15,16)(17,18,21,22)(19,24,23,20);(2,5)(3,7)(18,22)(19,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,23)(18,20,22,24);(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,22)(21,24);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=fg,e^2=gh,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 | C2×#39097, #7636480, #7636512, … |
| maximal normal subgroup of | C2×#39097, #7636480, #7636512, … |
| maximal quotient of | #521580, #522279, #522833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32085);