Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | Aut(G) order 8192 |
| 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 | #3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 327 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:
(1,2,6,12,7,13,21,11)(3,14,9,8,17,19,16,4)(5,20,22,10,23,26,32,27)(15,18,25,28,30,24,29,31);(1,3,6,9,7,17,21,16)(2,8,11,14,13,4,12,19)(5,15,22,25,23,30,32,29)(10,24,20,28,27,18,26,31);(1,4,7,8)(2,9)(3,12)(5,18,23,24)(6,19,21,14)(10,25)(11,17)(13,16)(15,26)(20,30)(22,31,32,28)(27,29);(1,5,21,32,7,23,6,22)(2,10,12,26,13,27,11,20)(3,15,16,29,17,30,9,25)(4,18,14,28,8,24,19,31)Polycyclic presentation:
2^8|a^2=egh,b^2=e,c^2=ef,d^2=eh,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 | #142696, #142698, #142700, … |
| maximal normal subgroup of | #142696, #142698, #142700, … |
| maximal quotient of | #316150, #316153, #316156, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20977);