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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,12)(8,13)(9,15)(10,11)(14,16)(20,28)(23,30)(26,32)(29,31)(33,34,39,42)(35,40,36,37)(38,46,47,41)(43,45,44,48);(2,4)(7,11)(8,12)(14,15)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,48,46,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,48,46,45);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,16,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,37,39,40)(34,36,42,35)(38,45,47,48)(41,44,46,43);(1,5,6,13)(2,8,4,12)(3,9,10,16)(7,14,11,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,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×#52858, #8092679, #8092704, … |
| maximal normal subgroup of | C2×#52858, #8092679, #8092704, … |
| maximal quotient of | #582302, #582515, #582559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45846);