Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(2,4)(5,6,10,15,11,17,27,16)(7,13,20,30,21,32,36,31)(8,22,24,29,25,28,19,12)(9,26,23,14,18,33,35,34);(1,2,3,4)(5,7,18,24,27,36,23,8,11,21,9,19,10,20,35,25)(6,12,14,13,15,22,33,30,17,29,34,32,16,28,26,31);(5,8,23,20,10,24,18,21,11,25,35,36,27,19,9,7)(6,13,26,12,16,31,34,28,17,32,33,29,15,30,14,22)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #93921, #93926, #152536, … |
| maximal normal subgroup of | #93921, #93926, #152536, … |
| maximal quotient of | #140725, #140726, #140727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6230);