Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2408 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2408 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
651 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)(3,5)(4,17)(6,25)(7,12)(8,13)(9,15)(10,21)(11,23)(14,26)(16,24)(18,19)(20,28)(22,27)(29,32)(30,31);(1,3,6,15,8,16,7,11)(2,9,12,5,13,23,25,24)(4,18,20,14,22,29,21,30)(10,19,27,31,28,32,17,26);(1,4)(2,10)(3,14)(5,19)(6,20)(7,21)(8,22)(9,26)(11,18)(12,27)(13,28)(15,29)(16,30)(17,25)(23,31)(24,32);(1,5,6,23,8,24,7,9)(2,11,12,3,13,15,25,16)(4,19,20,31,22,32,21,26)(10,18,27,14,28,29,17,30)Polycyclic presentation:
2^8|b^2=e,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24295, #6442699, #6442700, … |
| maximal normal subgroup of | C2×#24295, #6442699, #6442700, … |
| maximal quotient of | #263035, #263038, #263041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17283);