Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,11)(12,15,16,14)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(18,23)(19,24)(25,30)(26,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,16)(10,14,11,15)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32);(1,3,4,7)(2,6,5,8)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32);(1,2)(3,6)(4,5)(7,8)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#38498, #7636273, #7636312, … |
| maximal normal subgroup of | C2×#38498, #7636273, #7636312, … |
| maximal quotient of | #521819, #522393, #523041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31486);