Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,12)(11,15,16,14)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30);(1,2)(3,4)(5,6)(7,8)(9,10,13,12)(11,14,16,15)(18,23)(19,26)(24,30)(27,32);(2,6)(3,7)(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31);(1,3,5,7)(2,4,6,8)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(1,4)(2,7)(3,6)(5,8)(9,12,13,10)(11,15,16,14)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#42090, #7640431, #7640482, … |
| maximal normal subgroup of | C2×#42090, #7640431, #7640482, … |
| maximal quotient of | #544829, #545266, #545773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35078);