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 | |
1043 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)(3,5)(4,8)(6,7)(17,18)(19,22)(20,21)(23,24);(1,3)(2,5)(4,6)(7,8)(9,10,13,14)(11,16,15,12)(18,20)(22,23);(2,7)(3,4)(10,14)(11,15)(17,18,21,20)(19,23,24,22);(1,4,6,3)(2,8,7,5)(9,11,13,15)(10,12,14,16)(17,19,21,24)(18,22,20,23);(1,5,6,8)(2,3,7,4)(9,12)(10,15)(11,14)(13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|b^2=gh,c^2=f,d^2=h,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×#51133, #7707008, #7707054, … |
| maximal normal subgroup of | C2×#51133, #7707008, #7707054, … |
| maximal quotient of | #560871, #561121, #561975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44121);