Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
717 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 717 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,4)(5,6)(7,18)(8,13)(9,24)(10,16)(11,17)(12,20)(14,25)(15,22)(19,33)(21,28)(23,27)(26,30)(29,35)(31,36)(32,34);(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,25,34,26,23,36,35)(14,27,24,32,31,33,30,29);(1,2,3,4)(5,8)(6,13)(7,11)(9,23)(10,21)(12,15)(14,29)(16,28)(17,18)(19,26)(20,22)(24,27)(25,35)(30,33)(31,32)(34,36);(5,9,22,36,11,26,10,25)(6,14,16,30,17,31,15,24)(7,19,21,35,8,23,20,34)(12,27,13,29,28,33,18,32)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #501031, #501037, #501039, … |
| maximal normal subgroup of | #501031, #501037, #501039, … |
| maximal quotient of | #299255, #299259, #299262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18791);