Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2688 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,9,17,27)(4,18,5,21)(7,13,8,12)(10,22,28,20)(14,30,31,25)(15,24,19,32)(16,29,23,26);(1,3,8,16,6,17,7,23)(2,9,13,26,11,27,12,29)(4,19,22,14,5,15,20,31)(10,25,21,24,28,30,18,32);(1,4,6,5)(2,10,11,28)(3,14)(7,22,8,20)(9,24)(12,21,13,18)(15,23)(16,19)(17,31)(25,26)(27,32)(29,30);(1,5,6,4)(2,10,11,28)(3,15,17,19)(7,20,8,22)(9,25,27,30)(12,21,13,18)(14,23,31,16)(24,26,32,29)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6792411, #6792415, #6792416, … |
| maximal normal subgroup of | #6792411, #6792415, #6792416, … |
| maximal quotient of | #346001, #346003, #346004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24405);