Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 837 conjugacy classes of subgroups — too many to draw. Only its 239 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)(5,6,9,14,10,13,15,27)(7,11,18,23,17,24,21,28)(8,12,20,26,19,25,29,35)(16,22,31,34,30,33,32,36);(5,7,15,18,10,17,9,21)(6,11,14,28,13,24,27,23)(8,16,29,31,19,30,20,32)(12,22,26,36,25,33,35,34);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,22)(13,25)(14,26)(15,29)(17,30)(18,31)(21,32)(23,34)(24,33)(27,35)(28,36);(8,19)(12,25)(16,30)(20,29)(22,33)(26,35)(31,32)(34,36)Polycyclic presentation:
2^8|a^2=g,b^2=egh,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147503, #147514, #148024, … |
| maximal normal subgroup of | #147503, #147514, #148024, … |
| maximal quotient of | #195928, #195933, #195935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13979);