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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(12,15)(17,18,22,20)(19,24,21,23);(1,2)(3,7)(4,8)(5,6)(9,10,13,14)(11,12,16,15)(18,20)(23,24);(2,6)(3,8)(9,11,13,16)(10,15,14,12)(17,19,22,21)(18,23,20,24);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20,22,18)(19,24,21,23);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=g,e^2=g,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×#48637, #7703318, #7704119, … |
| maximal normal subgroup of | C2×#48637, #7703318, #7704119, … |
| maximal quotient of | #560637, #560967, #561745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41625);