Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,13)(11,15)(17,18,21,25)(19,23,27,22)(20,24,28,31)(26,30,32,29);(1,2)(3,6)(4,5)(7,8)(17,19,21,27)(18,22,25,23)(20,26,28,32)(24,29,31,30);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,24,25,31)(19,26,27,32)(22,29,23,30)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52403, #8092364, #8092438, … |
| maximal normal subgroup of | C2×#52403, #8092364, #8092438, … |
| maximal quotient of | #582247, #582467, #582469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45391);