Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| 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 | C2×#120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 871 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:
(2,4)(5,6)(7,11)(8,18,20,12)(9,21)(10,14)(13,22)(15,23,28,31)(16,29)(17,25)(19,26,33,32)(24,30)(27,35,36,34);(1,2,3,4)(5,7,10,17)(6,11,14,25)(8,15,20,28)(9,16,22,30)(12,23,18,31)(13,24,21,29)(19,27,33,36)(26,34,32,35);(5,8)(6,12)(7,15)(9,19)(10,20)(11,23)(13,26)(14,18)(16,27)(17,28)(21,32)(22,33)(24,34)(25,31)(29,35)(30,36);(5,9,10,22)(6,13,14,21)(7,16,17,30)(8,19,20,33)(11,24,25,29)(12,26,18,32)(15,27,28,36)(23,34,31,35)Polycyclic presentation:
2^8|a^2=g,b^2=egh,d^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,f^b=fh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147800, #147801, #147838, … |
| maximal normal subgroup of | #147800, #147801, #147838, … |
| maximal quotient of | #193867, #193873, #193879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13950);