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:
(1,2,5,8)(3,9,10,6)(4,7,12,14)(11,15,16,13)(17,18,21,22)(19,23,25,29)(20,24,27,30)(26,31,28,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(3,10)(6,9)(11,16)(13,15)(18,22)(19,25)(20,27)(31,32)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,28)(21,25)(22,29)(24,32)(26,27)(30,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,27)(22,30)(23,31)(25,28)(29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,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×#57814, #8101833, #8101965, … |
| maximal normal subgroup of | C2×#57814, #8101833, #8101965, … |
| maximal quotient of | #586358, #586461, #587288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50802);