Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1005 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)(3,7)(4,5)(6,8)(17,18,22,24)(19,21,20,23);(2,5)(6,7)(9,10,12,13)(11,15,16,14)(17,19,22,20)(18,23,24,21);(10,13)(14,15)(18,24)(19,20);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,24,21);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,19,24,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,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 | #133502, #500042, #500390, … |
| maximal normal subgroup of | #133502, #500042, #500390, … |
| maximal quotient of | #561217, #562418, #564621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44918);