Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C2×#72232847 (order 4096) · Inn(G) = C25 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,16,25,29)(11,15,21,28,19,26,33,37)(12,22,23,36,24,17,30,31)(20,34,35,40,32,27,38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,28)(22,34)(23,35)(24,32)(25,33)(29,37)(30,38)(31,39)(36,40);(3,8)(5,7)(10,16)(11,19)(17,22)(18,29)(20,32)(21,33)(31,36)(35,38);(2,6)(5,7)(11,19)(15,26)(20,32)(21,33)(27,34)(28,37)(35,38)(39,40);(9,12,13,23,14,24,25,30)(10,17,18,31,16,22,29,36)(11,20,21,35,19,32,33,38)(15,27,28,39,26,34,37,40)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,e^a=eh,c^b=cfh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129614, #495780, #511947, … |
| maximal normal subgroup of | #129614, #495780, #511947, … |
| maximal quotient of | #6370465, #6370471, #6370472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53303);