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:
(2,6)(3,9)(8,13)(10,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,41)(36,38)(37,44)(39,42)(40,45)(43,46)(47,48);(2,6)(7,12)(8,13)(14,16)(18,19)(20,27)(21,29)(30,31)(34,36)(35,42)(40,44)(43,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,34)(35,41,42,39)(37,44,45,40)(43,47,48,46);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,36,44)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^8|a^2=gh,c^2=fg,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×#54520, #8102735, #8103195, … |
| maximal normal subgroup of | C2×#54520, #8102735, #8103195, … |
| maximal quotient of | #586466, #586944, #586982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47508);