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 | |
613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,25)(20,22)(21,28)(23,27)(24,29)(26,30)(31,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(2,3)(4,11)(5,13)(14,15)(18,20)(21,29)(23,25)(26,32)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,46)(34,40,41,45)(35,43,36,44)(39,47,42,48)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=f,d^2=g,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×#56181, #8102830, #8103194, … |
| maximal normal subgroup of | C2×#56181, #8102830, #8103194, … |
| maximal quotient of | #585793, #586481, #586962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49169);