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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,35)(36,43)(37,45)(46,47);(4,12)(7,14)(10,16)(13,15)(18,22)(19,25)(20,29)(30,32)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^8|a^2=fgh,b^2=fh,c^2=f,d^2=g,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×#56143, #8103020, #8103193, … |
| maximal normal subgroup of | C2×#56143, #8103020, #8103193, … |
| maximal quotient of | #586401, #586643, #586976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49131);