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 | |
636 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)(11,15)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(4,12)(8,13)(11,15)(14,16)(18,19)(20,27)(24,26)(28,32)(33,35,38,36)(34,39,42,40)(37,44,46,43)(41,48,45,47);(1,2,5,6)(3,10,9,7)(4,8,12,13)(11,16,15,14)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=g,d^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×#55840, #8101701, #8101967, … |
| maximal normal subgroup of | C2×#55840, #8101701, #8101967, … |
| maximal quotient of | #585849, #586880, #587238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48828);