Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 495 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:
(1,2,4,3)(5,6)(7,16)(8,20)(9,14)(10,25)(11,15)(12,18)(13,23)(17,31)(19,27)(21,35)(22,29)(24,28)(26,36)(30,33)(32,34);(1,3,4,2)(5,7,10,18,11,19,9,24)(6,12,14,16,15,28,25,27)(8,17,22,32,23,33,21,36)(13,26,29,30,20,34,35,31);(5,8,10,22,11,23,9,21)(6,13,14,29,15,20,25,35)(7,17,18,32,19,33,24,36)(12,26,16,30,28,34,27,31);(5,9,11,10)(6,14,15,25)(7,18,19,24)(8,21,23,22)(12,27,28,16)(13,29,20,35)(17,32,33,36)(26,31,34,30)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #360291, #367942, #368475, … |
| maximal normal subgroup of | #360291, #367942, #368475, … |
| maximal quotient of | #202480, #202486, #202492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14065);