Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
737 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 737 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,16)(8,20)(9,14)(10,24)(11,15)(12,18)(13,23)(17,34)(19,28)(21,27)(22,32)(25,29)(26,31)(30,35)(33,36);(1,2,4,3)(5,7,9,18)(6,12,14,16)(8,21,22,36)(10,19,11,25)(13,30,31,34)(15,28,24,29)(17,26,35,23)(20,33,32,27);(5,8,10,23,9,22,11,26)(6,13,15,20,14,31,24,32)(7,17,19,36,18,35,25,21)(12,27,28,30,16,33,29,34);(1,3,4,2)(6,14)(8,22)(12,16)(15,24)(17,35)(21,36)(23,26)(28,29)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152736, #152737, #152740, … |
| maximal normal subgroup of | #152736, #152737, #152740, … |
| maximal quotient of | #297411, #297413, #297419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18739);