Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 167 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)(5,6,11,16,12,17,10,15)(7,18,22,13,23,30,21,31)(8,24,25,33,26,32,9,14)(19,35,27,34,20,28,36,29);(1,3,4,2)(5,7,12,23)(6,13,17,31)(8,19,26,20)(9,27,25,36)(10,22,11,21)(14,28,33,35)(15,30,16,18)(24,34,32,29);(5,8,10,9,12,26,11,25)(6,14,16,24,17,33,15,32)(7,19,22,27,23,20,21,36)(13,28,18,34,31,35,30,29);(5,9,10,26,12,25,11,8)(6,14,15,32,17,33,16,24)(7,20,21,27,23,19,22,36)(13,29,18,28,31,34,30,35)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=f,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152475, #152576, #152578, … |
| maximal normal subgroup of | #152475, #152576, #152578, … |
| maximal quotient of | #231471, #231478, #231480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16191);