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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,22)(21,24);(2,5)(3,7)(9,11,13,12)(10,14,16,15)(17,18,20,22)(19,21,24,23);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24);(17,19)(18,21)(20,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=fh,e^2=g,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×#38235, #7635482, #7635654, … |
| maximal normal subgroup of | C2×#38235, #7635482, #7635654, … |
| maximal quotient of | #521691, #522308, #522825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31223);