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×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 502 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)(5,6,9,15,11,17,10,16)(7,18,20,30,22,29,21,12)(8,13,24,14,25,32,26,33)(19,31,23,34,35,28,36,27);(5,7,11,22)(6,12,17,30)(8,23,25,36)(9,21,10,20)(13,31,32,28)(14,27,33,34)(15,29,16,18)(19,26,35,24);(2,4)(5,8)(6,13)(7,19)(9,24)(10,26)(11,25)(12,27)(14,15)(16,33)(17,32)(18,31)(20,23)(21,36)(22,35)(28,29)(30,34);(2,4)(5,8)(6,14)(7,19)(9,26)(10,24)(11,25)(12,28)(13,15)(16,32)(17,33)(18,34)(20,36)(21,23)(22,35)(27,29)(30,31)Polycyclic presentation:
2^8|a^2=e,b^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 | #153161, #153162, #153168, … |
| maximal normal subgroup of | #153161, #153162, #153168, … |
| maximal quotient of | #332984, #332990, #332997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23479);