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 | |
1016 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)(3,4)(5,6)(7,8)(17,18)(19,23)(20,21)(22,24);(2,6)(3,7)(9,10,12,13)(11,15,16,14)(18,21)(22,23);(10,13)(14,15)(17,19)(18,22)(20,24)(21,23);(1,3,5,7)(2,4,6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(1,4)(2,7)(3,6)(5,8)(9,11,12,16)(10,14,13,15)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^8|b^2=h,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×#51926, #7707908, #7707915, … |
| maximal normal subgroup of | C2×#51926, #7707908, #7707915, … |
| maximal quotient of | #561223, #561313, #562423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44914);