Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
647 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 135 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)(7,13)(8,21)(9,24)(10,17)(11,16)(12,19)(14,25)(15,22)(18,32)(20,28)(23,27)(26,31)(29,33)(30,36)(34,35);(5,7,10,19,11,8,22,20)(6,12,15,13,16,28,17,21)(9,18,25,33,26,23,36,34)(14,27,24,29,30,32,31,35);(2,4)(5,8)(6,13)(7,11)(9,23)(10,20)(12,17)(14,29)(15,28)(16,21)(18,26)(19,22)(24,32)(25,34)(27,31)(30,35)(33,36);(5,9,22,36,11,26,10,25)(6,14,17,31,16,30,15,24)(7,18,20,34,8,23,19,33)(12,27,21,35,28,32,13,29)Polycyclic presentation:
2^8|b^2=e,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 | #499770, #499772, #499773, … |
| maximal normal subgroup of | #499770, #499772, #499773, … |
| maximal quotient of | #263036, #263039, #263042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17278);