Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(19,23)(22,24);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,16,15,12)(17,18,21,20)(19,22,23,24);(3,8)(6,7)(10,14)(11,15)(17,19,21,23)(18,22,20,24);(1,3)(2,7)(4,6)(5,8)(9,11,13,15)(10,12,14,16)(17,20,21,18)(19,24,23,22);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,14)(13,16)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50318, #7705993, #7706220, … |
| maximal normal subgroup of | C2×#50318, #7705993, #7706220, … |
| maximal quotient of | #561273, #561398, #562302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43306);