Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 663 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)(3,4)(5,6)(7,15)(8,19)(9,23)(10,13)(11,17)(12,22)(14,25)(16,32)(18,30)(20,31)(21,29)(24,27)(26,35)(28,33)(34,36);(5,7,10,18)(6,11,13,27)(8,20,22,16)(9,24,25,17)(12,28,19,26)(14,30,23,15)(21,33,36,35)(29,31,34,32);(2,4)(5,8,9,21,10,22,25,36)(6,12,14,29,13,19,23,34)(7,16,17,33,18,20,24,35)(11,26,15,31,27,28,30,32);(6,13)(11,27)(12,19)(14,23)(15,30)(26,28)(29,34)(31,32)Polycyclic presentation:
2^8|b^2=h,c^2=e,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,c^b=cg,e^b=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #376490, #389324, #389327, … |
| maximal normal subgroup of | #376490, #389324, #389327, … |
| maximal quotient of | #238844, #238846, #238848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16510);