Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,13,10,14,23,30)(7,11,16,25,17,26,34,36)(8,18,20,28,21,29,22,12)(15,31,32,27,19,35,33,24);(1,2,4,3)(5,7,9,16,10,17,23,34)(6,11,13,25,14,26,30,36)(8,19,20,33,21,15,22,32)(12,27,18,35,28,24,29,31);(5,8,10,21)(6,12,14,28)(7,15,17,19)(9,22,23,20)(11,24,26,27)(13,29,30,18)(16,33,34,32)(25,35,36,31);(1,3,4,2)Polycyclic presentation:
2^8|a^2=f,b^2=ef,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #285123, #285125, #285140, … |
| maximal normal subgroup of | #285123, #285125, #285140, … |
| maximal quotient of | #287741, #287748, #287753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18394);