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 | C24 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
859 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 859 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)(7,11,17,24)(8,12)(9,13)(10,14)(15,22,30,34)(16,23,32,36)(18,25)(19,26)(20,27)(21,28)(29,33,31,35);(5,7)(6,11,14,24)(8,15)(9,16)(10,17)(12,22,26,34)(13,23,28,36)(18,29)(19,30)(20,31)(21,32)(25,33,27,35);(2,4)(5,8,10,19)(6,12,14,26)(7,15,17,30)(9,20,21,18)(11,22,24,34)(13,27,28,25)(16,31,32,29)(23,35,36,33);(5,9,10,21)(6,13,14,28)(7,16,17,32)(8,18,19,20)(11,23,24,36)(12,25,26,27)(15,29,30,31)(22,33,34,35)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,d^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 | #147515, #148012, #364605, … |
| maximal normal subgroup of | #147515, #148012, #364605, … |
| maximal quotient of | #195930, #195934, #195935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13974);