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:
(1,2,6,4)(3,7,11,9)(5,12,13,8)(10,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(2,4)(7,9)(8,12)(14,15)(18,21)(19,25)(20,28)(31,32)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,22,25)(18,23,21,27)(20,29,28,26)(24,32,30,31)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,28)(18,24,21,30)(19,26,25,29)(23,31,27,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21,22,18)(19,27,25,23)(20,30,28,24)(26,32,29,31)(33,37,38,45)(34,40,41,48)(35,42,36,44)(39,46,43,47)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=fg,d^2=f,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×#54599, #8103684, #8103904, … |
| maximal normal subgroup of | C2×#54599, #8103684, #8103904, … |
| maximal quotient of | #586629, #586922, #587292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47587);