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:
(2,6)(3,9)(8,13)(10,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,6)(7,12)(8,13)(14,16)(18,22)(19,25)(20,29)(30,32)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)Polycyclic presentation:
2^8|a^2=fgh,b^2=fgh,c^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×#55991, #8103087, #8103198, … |
| maximal normal subgroup of | C2×#55991, #8103087, #8103198, … |
| maximal quotient of | #586644, #586811, #587289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 48979);