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 | #2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,15)(11,16,13,12)(17,18,21,23)(19,20,24,22);(1,2,4,5)(3,7,8,6)(10,15)(11,13)(17,19,21,24)(18,22,23,20);(2,5)(6,7)(9,11,14,13)(10,16,15,12)(17,20)(18,19)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,13)(17,19)(18,20)(21,24)(22,23);(1,3,4,8)(2,6,5,7)(9,13,14,11)(10,16,15,12)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=fg,d^2=fg,e^2=h,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×#51955, #7707884, #7707955, … |
| maximal normal subgroup of | C2×#51955, #7707884, #7707955, … |
| maximal quotient of | #561219, #561319, #562292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44943);