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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
629 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)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(33,34,39,42)(35,40,36,37)(38,41,46,48)(43,47,44,45);(2,4)(3,9)(5,13)(14,16)(18,23)(20,24)(26,29)(28,31)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fgh,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×#56832, #8100133, #8100300, … |
| maximal normal subgroup of | C2×#56832, #8100133, #8100300, … |
| maximal quotient of | #586269, #586541, #587165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49820);