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 | #2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
780 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)(9,10,12,14)(11,15,16,13)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,16,15)(18,19)(20,27)(24,26)(28,32);(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31);(1,3)(2,6)(4,7)(5,8)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=f,d^2=fg,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×#52544, #8092310, #8092467, … |
| maximal normal subgroup of | C2×#52544, #8092310, #8092467, … |
| maximal quotient of | #582342, #582528, #582537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45532);