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 | |
898 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,23,24,20);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,16)(14,15)(18,22)(20,23);(3,4)(7,8)(10,14)(11,16)(17,19)(18,20)(21,24)(22,23);(1,4,6,3)(2,7,5,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,23)(22,24);(1,5,6,2)(3,7,4,8)(9,12,13,15)(10,11,14,16)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^8|a^2=fh,b^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×#51833, #7707838, #7707851, … |
| maximal normal subgroup of | C2×#51833, #7707838, #7707851, … |
| maximal quotient of | #561301, #561318, #562454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44821);