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 | |
651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18,22,25)(19,21,24,30)(20,23,29,32)(26,28,27,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,6)(3,10)(8,12)(11,16)(19,24)(21,30)(26,27)(28,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,22,24)(18,21,25,30)(20,27,29,26)(23,31,32,28)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,24)(19,25)(20,28)(22,30)(23,27)(26,32)(29,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fgh,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×#56971, #8099538, #8100314, … |
| maximal normal subgroup of | C2×#56971, #8099538, #8100314, … |
| maximal quotient of | #586414, #586480, #587125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49959);