Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13497 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2075 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2075 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,6,12,5,11,21,27,8,13,24,29,7,14,23,30)(3,15,17,31,20,9,26,22,10,4,18,19,16,25,32,28);(1,3)(2,9)(4,11)(5,20)(6,18)(7,16)(8,10)(12,28)(13,25)(14,15)(17,24)(19,30)(21,32)(22,29)(23,26)(27,31);(1,4,5,9,8,15,7,25)(2,10,11,20,13,3,14,16)(6,22,21,31,24,28,23,19)(12,26,27,17,29,32,30,18)Polycyclic presentation:
2^8|a^2=e,c^2=d,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13497, #499384, #499603, … |
| maximal normal subgroup of | C2×#13497, #499384, #499603, … |
| maximal quotient of | #146013, #146032, #146053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6485);