Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(11,15)(14,16)(18,22)(20,23);(2,5)(3,7)(10,13)(11,15)(17,18,21,22)(19,23,24,20);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,23)(22,24);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,22)(23,24)Polycyclic presentation:
2^8|b^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46107, #7702596, #7702715, … |
| maximal normal subgroup of | C2×#46107, #7702596, #7702715, … |
| maximal quotient of | #557289, #557330, #557496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39095);