Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,22,23)(19,21,20,24);(2,3)(4,8)(9,11)(10,15)(12,14)(13,16)(18,23)(19,20);(2,3)(6,7)(11,12)(15,16)(17,19,22,20)(18,24,23,21);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,13,16)(17,20,22,19)(18,21,23,24);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,15,12,16)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^8|a^2=fgh,c^2=fgh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#56951, #8099632, #8100317, … |
| maximal normal subgroup of | C2×#56951, #8099632, #8100317, … |
| maximal quotient of | #585247, #585501, #587143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49939);