Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(3,12)(4,17)(5,6)(8,22)(9,24)(10,21)(13,25)(14,29)(15,30)(16,18)(19,20)(23,32)(26,31)(27,28);(1,2)(3,13)(4,9)(5,21)(6,10)(7,11)(8,15)(12,28)(14,27)(16,23)(17,24)(18,22)(19,31)(20,26)(25,29)(30,32);(1,3,6,15,7,14,5,16)(2,8,10,27,11,23,21,13)(4,18,17,29,20,30,19,12)(9,25,24,32,26,28,31,22)Polycyclic presentation:
2^8|c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,f^b=fh,d^c=dh,e^c=egh,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13396, #499050, #499269, … |
| maximal normal subgroup of | C2×#13396, #499050, #499269, … |
| maximal quotient of | #144716, #144752, #145024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6384);