Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29);(2,7)(6,9)(8,14)(13,15)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=h,f^2=g,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #106672, #106791, #106792, … |
| maximal normal subgroup of | #106672, #106791, #106792, … |
| maximal quotient of | #95154, #95155, #95159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1108);