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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
663 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,18,21,22)(19,23,25,29)(20,24,27,30)(26,31,28,32)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(2,6)(7,13)(8,11)(14,15)(18,22)(19,25)(20,27)(31,32)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,19)(18,23)(20,28)(21,25)(22,29)(24,32)(26,27)(30,31)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,20)(18,24)(19,26)(21,27)(22,30)(23,31)(25,28)(29,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(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=g,c^2=fgh,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×#56981, #8100133, #8100315, … |
| maximal normal subgroup of | C2×#56981, #8100133, #8100315, … |
| maximal quotient of | #586541, #587293, #587301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49969);