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 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10,12,14)(11,15,16,13)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31);(2,3)(4,8)(18,19)(21,29)(23,25)(27,32);(1,2,5,3)(4,6,8,7)(9,11,12,16)(10,13,14,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,5,2)(4,7,8,6)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4)(2,6)(3,7)(5,8)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,c^2=fg,d^2=h,e^2=f,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×#46151, #7702796, #7702839, … |
| maximal normal subgroup of | C2×#46151, #7702796, #7702839, … |
| maximal quotient of | #557114, #557119, #557405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39139);