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 | |
652 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)(3,9)(4,6)(5,12)(7,10)(8,13)(11,14)(15,16)(17,18,22,23)(19,21,27,24)(20,28,29,25)(26,32,30,31)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(2,4)(3,10)(8,12)(11,16)(18,23)(20,29)(21,24)(26,30)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,19)(18,24)(20,30)(21,23)(22,27)(25,31)(26,29)(28,32)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20,22,29)(18,25,23,28)(19,26,27,30)(21,31,24,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,24)(18,19,23,27)(20,31,29,32)(25,26,28,30)(33,37,38,45)(34,40,41,48)(35,42,36,44)(39,46,43,47)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,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×#54584, #8103065, #8103195, … |
| maximal normal subgroup of | C2×#54584, #8103065, #8103195, … |
| maximal quotient of | #585729, #586545, #586954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47572);