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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,17,14,18,27,34)(11,15,20,29,21,30,38,40)(12,22,24,32,25,33,26,16)(19,35,36,31,23,39,37,28);(2,5)(6,7)(9,11,13,20,14,21,27,38)(10,15,17,29,18,30,34,40)(12,23,24,37,25,19,26,36)(16,31,22,39,32,28,33,35);(9,12,14,25)(10,16,18,32)(11,19,21,23)(13,26,27,24)(15,28,30,31)(17,33,34,22)(20,37,38,36)(29,39,40,35);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=f,b^2=f,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 | #285124, #285126, #285139, … |
| maximal normal subgroup of | #285124, #285126, #285139, … |
| maximal quotient of | #286653, #286660, #286665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18302);