Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
664 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 664 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(5,6)(7,19)(8,23)(9,13)(10,17)(11,16)(12,18)(14,25)(15,22)(20,34)(21,29)(24,33)(26,32)(27,31)(28,35)(30,36);(1,2)(3,4)(5,7,11,21,12,22,10,9)(6,13,17,15,18,29,16,19)(8,20,27,36,26,35,25,24)(14,28,32,30,31,34,23,33);(2,4)(5,8)(6,14)(7,20)(9,24)(10,25)(11,27)(12,26)(13,28)(15,30)(16,23)(17,32)(18,31)(19,33)(21,36)(22,35)(29,34);(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,29)(8,24,25,35,26,36,27,20)(14,30,23,28,31,33,32,34)Polycyclic presentation:
2^8|b^2=eh,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139570, #150465, #486312, … |
| maximal normal subgroup of | #139570, #150465, #486312, … |
| maximal quotient of | #262676, #262681, #262685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17267);