Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 199 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)(7,19)(8,23)(9,15)(10,17)(11,16)(12,18)(13,21)(14,24)(20,34)(22,32)(25,33)(26,36)(27,30)(28,31)(29,35);(5,7,12,22)(6,13,18,31)(8,9,25,27)(10,28,11,21)(14,30,34,15)(16,32,17,19)(20,29,24,26)(23,36,33,35);(5,8,11,24,12,25,10,20)(6,14,17,23,18,34,16,33)(7,9,21,26,22,27,28,29)(13,30,19,36,31,15,32,35);(5,9)(6,15)(7,20)(8,21)(10,29)(11,26)(12,27)(13,23)(14,32)(16,36)(17,35)(18,30)(19,34)(22,24)(25,28)(31,33)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136391, #136421, #136864, … |
| maximal normal subgroup of | #136391, #136421, #136864, … |
| maximal quotient of | #225495, #225500, #225505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15678);