Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10326 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
300 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 300 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,7,11,16)(4,6,12,8)(10,15,13,14)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(17,20,21,27)(18,23,24,31)(19,25,26,32)(22,28,29,30)Polycyclic presentation:
2^8|a^2=g,c^2=gh,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,d^c=dg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10332, #234360, #234362, … |
| maximal normal subgroup of | C2×#10332, #234360, #234362, … |
| maximal quotient of | #102621, #102622, #102623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3320);