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 | |
615 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)(3,9)(4,6)(5,12)(7,11)(8,13)(10,14)(15,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(2,4)(5,13)(7,9)(10,16)(21,29)(24,31)(26,32)(28,30)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4,6,2)(3,7,11,9)(5,12,13,8)(10,14,16,15)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^8|a^2=fgh,b^2=h,c^2=gh,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×#57931, #8101568, #8101956, … |
| maximal normal subgroup of | C2×#57931, #8101568, #8101956, … |
| maximal quotient of | #585756, #586360, #586724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50919);