Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2504 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2504 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,6,27,8,13,7,12)(3,14,16,9,17,20,28,29)(4,18,22,30,23,32,21,10)(5,15,24,11,26,19,25,31);(1,3,6,16,8,17,7,28)(2,9,12,14,13,29,27,20)(4,19,21,11,23,15,22,31)(5,10,24,18,26,30,25,32);(1,4)(2,10,13,30)(3,15,17,19)(5,20)(6,21)(7,22)(8,23)(9,24)(11,16,31,28)(12,18,27,32)(14,26)(25,29);(1,5,6,24,8,26,7,25)(2,11,12,15,13,31,27,19)(3,10,16,18,17,30,28,32)(4,20,21,9,23,14,22,29)Polycyclic presentation:
2^8|a^2=eg,b^2=e,c^2=ef,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #503492, #503493, #503497, … |
| maximal normal subgroup of | #503492, #503493, #503497, … |
| maximal quotient of | #316149, #316151, #316154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20975);