Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(2,4)(3,9)(5,13)(14,16)(19,26)(22,25)(27,32)(29,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57771, #8101568, #8101963, … |
| maximal normal subgroup of | C2×#57771, #8101568, #8101963, … |
| maximal quotient of | #585722, #586468, #586932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50759);