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 | |
611 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,7,11,9)(5,12,13,8)(10,15,16,14)(17,18)(19,24)(20,26)(21,23)(22,25)(27,31)(28,32)(29,30)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,4)(7,9)(8,12)(14,15)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=fg,d^2=fg,e^2=f,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×#55905, #8103156, #8103186, … |
| maximal normal subgroup of | C2×#55905, #8103156, #8103186, … |
| maximal quotient of | #585954, #586045, #587298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48893);