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:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,48)(44,45);(2,3)(4,11)(5,13)(14,15)(19,26)(22,25)(27,32)(29,31)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,48,46,45);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,48,46,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,37,39,40)(34,35,42,36)(38,45,47,48)(41,43,46,44);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,38,39,47)(34,41,42,46)(35,43,36,44)(37,45,40,48)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=f,d^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×#58545, #8104264, #8104430, … |
| maximal normal subgroup of | C2×#58545, #8104264, #8104430, … |
| maximal quotient of | #586693, #586930, #587164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51533);