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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(2,6)(3,10)(8,12)(11,16)(18,22)(23,29)(24,27)(30,31)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=fg,d^2=g,e^2=g,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×#56243, #8099917, #8100310, … |
| maximal normal subgroup of | C2×#56243, #8099917, #8100310, … |
| maximal quotient of | #586633, #586979, #587275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49231);