Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
498 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2)(3,4)(5,6,11,12,10,16,9,15)(7,17,25,28,20,13,18,26)(8,21,24,14,23,30,22,29)(19,33,32,31,35,36,34,27);(6,12)(7,18)(8,22)(9,11)(14,29)(15,16)(20,25)(23,24)(26,28)(27,36)(31,33)(32,34);(2,4)(5,7,10,20)(6,13,16,17)(8,19,23,35)(9,18,11,25)(12,26,15,28)(14,27,29,31)(21,36,30,33)(22,34,24,32);(2,4)(5,8)(6,14)(7,19)(9,24)(10,23)(11,22)(12,21)(13,27)(15,30)(16,29)(17,31)(18,32)(20,35)(25,34)(26,36)(28,33)Polycyclic presentation:
2^8|a^2=eh,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=gh,g^c=gh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153165, #153166, #153170, … |
| maximal normal subgroup of | #153165, #153166, #153170, … |
| maximal quotient of | #332987, #332993, #333000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23482);