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 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(11,15)(14,16)(17,18,23,21)(19,26,24,20)(22,29,30,25)(27,28,31,32);(9,10)(11,14)(12,13)(15,16)(17,19,23,24)(18,20,21,26)(22,27,30,31)(25,28,29,32);(2,6)(3,8)(10,13)(14,16)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,21,23,18)(19,20,24,26)(22,29,30,25)(27,28,31,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^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×#52322, #8092253, #8092460, … |
| maximal normal subgroup of | C2×#52322, #8092253, #8092460, … |
| maximal quotient of | #582173, #582386, #582393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45310);