Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2410 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,19,28,20,30,18,9)(4,21,24,11,5,25,23,10)(16,26,22,29,31,32,17,27);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,29)(11,27)(12,30)(13,15)(14,28)(16,23)(21,32)(24,31)(25,26);(1,4,8,5)(2,10,14,11)(3,16,20,31)(6,23,7,24)(9,26,28,32)(12,21,13,25)(15,27,30,29)(17,18,22,19);(1,5,8,4)(2,11,14,10)(3,17,20,22)(6,23,7,24)(9,27,28,29)(12,21,13,25)(15,26,30,32)(16,18,31,19)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24475, #6437949, #6437950, … |
| maximal normal subgroup of | C2×#24475, #6437949, #6437950, … |
| maximal quotient of | #268284, #268286, #268295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17463);