Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(5,6)(7,17)(8,22,24,29)(9,15)(10,14)(11,16)(12,19)(13,23,30,25)(18,26,33,36)(20,28)(21,27)(31,32,35,34);(1,2,3,4)(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,17)(8,18,25,34,24,33,23,32)(13,26,22,35,30,36,29,31);(5,8,9,23,11,24,10,25)(6,13,14,29,16,30,15,22)(7,18,19,32,21,33,20,34)(12,26,17,31,28,36,27,35)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93359, #93553, #128806, … |
| maximal normal subgroup of | #93359, #93553, #128806, … |
| maximal quotient of | #108492, #108493, #108496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3802);