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 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(11,15)(14,16)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31);(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31);(2,6)(3,8)(10,13)(14,16)(17,20,23,19)(18,25,22,24)(21,28,30,27)(26,32,29,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,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×#52330, #8092391, #8092459, … |
| maximal normal subgroup of | C2×#52330, #8092391, #8092459, … |
| maximal quotient of | #582207, #582413, #582429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45318);