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 | |
755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(14,16)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28);(1,2,4,6)(3,7,8,5)(9,10,13,12)(11,14,15,16)(18,23)(19,24)(25,30)(27,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32);(9,12,13,10)(11,16,15,14)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32);(1,3,4,8)(2,5,6,7)(17,21,22,29)(18,25,23,30)(19,27,24,31)(20,28,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=f,d^2=fg,e^2=h,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×#39235, #7636961, #7637033, … |
| maximal normal subgroup of | C2×#39235, #7636961, #7637033, … |
| maximal quotient of | #521642, #522276, #522947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32223);