Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7310 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1304 |
| commutator | C24 | C42 |
| Frattini, prime radical, cosocle | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 37 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 — 2 generators on 32 points:
(1,2)(3,11,13,22)(4,15,16,8)(5,18,19,9)(6,10)(7,12)(14,29)(17,23)(20,26,31,32)(21,27)(24,30)(25,28);(1,3,5,14,6,13,19,28)(2,7,9,20,10,21,18,31)(4,12,17,26,16,27,30,32)(8,22,23,29,15,11,24,25)Polycyclic presentation:
2^8|a^2=d,b^2=e,e^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gh,c^b=cg,d^b=df,f^b=fh,d^c=dh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7312, #128040, #128077, … |
| maximal normal subgroup of | C2×#7312, #128040, #128077, … |
| maximal quotient of | #93101, #93102, #93103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 300);