Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1007 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 479 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) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,23,32)(12,26,27,34)(13,19,14,18)(16,25,31,24)(17,28,33,29)(22,30,36,40)(35,37,39,38);(2,5)(6,8)(9,11,14,25,15,23,13,24)(10,16,19,32,20,31,18,21)(12,22,29,38,27,36,28,37)(17,30,26,39,33,40,34,35);(1,3,4,7)(2,6,5,8)(9,12,13,28,15,27,14,29)(10,17,18,34,20,33,19,26)(11,22,24,37,23,36,25,38)(16,30,21,35,31,40,32,39);(12,27)(17,33)(22,36)(26,34)(28,29)(30,40)(35,39)(37,38);(11,23)(16,31)(21,32)(22,36)(24,25)(30,40)(35,39)(37,38)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=f,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #152070, #152072, #246932, … |
| maximal normal subgroup of | #152070, #152072, #246932, … |
| maximal quotient of | #6371047, #6371694, #6371695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53608);