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 | |
647 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)(19,25)(20,22)(21,28)(23,27)(24,29)(26,30)(31,32)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,48)(44,45);(2,3)(4,11)(5,13)(14,15)(18,20)(21,29)(23,25)(26,32)(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)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(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)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(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)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(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=fh,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×#58464, #8104210, #8104286, … |
| maximal normal subgroup of | C2×#58464, #8104210, #8104286, … |
| maximal quotient of | #585726, #586460, #586924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51452);