Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,8,13)(7,17,20,35,21,32,18,34)(9,23,22,33,25,14,26,31)(12,27,28,24,29,36,30,19);(5,7,11,21)(6,12,16,29)(8,20,10,18)(9,24,25,19)(13,30,15,28)(14,32,23,17)(22,27,26,36)(31,34,33,35);(2,4)(5,8,11,10)(6,13,16,15)(7,18,21,20)(9,22,25,26)(12,28,29,30)(14,31,23,33)(17,34,32,35)(19,27,24,36)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,d^b=dh,f^b=fgh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94421, #94443, #133352, … |
| maximal normal subgroup of | #94421, #94443, #133352, … |
| maximal quotient of | #129220, #129253, #129342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5692);