Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | Aut(G) = C22×#72231761 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| 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 | C25 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
327 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,5,10)(3,12,14,24)(4,16,19,8)(6,11,20,9)(7,17,22,13)(15,28,31,27)(18,26,29,25)(21,30,23,32);(1,3,6,15)(2,7,11,23)(4,17,18,32)(5,14,20,31)(8,24,25,27)(9,21,10,22)(12,26,28,16)(13,29,30,19);(4,18)(7,21)(8,25)(12,24)(13,30)(16,26)(17,32)(19,29)(22,23)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9944, #234928, #234929, … |
| maximal normal subgroup of | C2×#9944, #234928, #234929, … |
| maximal quotient of | #102986, #102989, #102992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2932);