Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,19,22,25,21,27)(20,24,29,26,30,32,31,28);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,14,15,16)(17,19,23,25,22,27,21,18)(20,28,29,24,30,26,31,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,30)(18,24,25,32)(19,26,27,28)(21,31,23,29);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,28,32,26)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=de,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23261, #6383179, #6383183, … |
| maximal normal subgroup of | C2×#23261, #6383179, #6383183, … |
| maximal quotient of | #356396, #356914, #356937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16249);