Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
261 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,11,23,30,4,20)(3,14,17,9,25,27,15,31)(6,12,21,8,7,13,24,10)(16,29,18,26,22,28,19,32);(2,8,12,11)(4,21)(5,24)(9,14,28,26)(10,13,20,30)(15,22)(16,17)(27,29,32,31);(1,3,7,19)(2,9,13,29)(4,22,21,15)(5,16,24,17)(6,18,23,25)(8,14,20,32)(10,27,11,26)(12,28,30,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9276, #198713, #198716, … |
| maximal normal subgroup of | C2×#9276, #198713, #198716, … |
| maximal quotient of | #99916, #99918, #99924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2264);