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 | |
909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(17,18,21,22)(19,20,23,24);(2,3)(6,7)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,24,23,20);(10,11)(14,15)(18,22)(19,23);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,20,22,24);(1,4,5,8)(2,6,3,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^8|a^2=fg,b^2=g,d^2=fh,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×#50680, #7706483, #7706494, … |
| maximal normal subgroup of | C2×#50680, #7706483, #7706494, … |
| maximal quotient of | #560455, #560633, #561466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43668);