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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(11,15)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34)(35,36)(37,45)(38,42)(39,40)(41,46)(43,47)(44,48);(4,12)(8,13)(11,15)(14,16)(19,26)(22,25)(27,32)(29,31)(33,34,38,42)(35,39,40,36)(37,45,46,41)(43,44,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,45)(46,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=f,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×#56845, #8100074, #8100310, … |
| maximal normal subgroup of | C2×#56845, #8100074, #8100310, … |
| maximal quotient of | #585772, #587011, #587231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49833);