Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
319 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,9)(3,10,13,14)(4,5,6,15)(8,16,12,11)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30);(1,3)(2,8)(4,11)(5,14)(6,16)(7,13)(9,12)(10,15)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,4,7,6)(2,5,9,15)(3,11,13,16)(8,14,12,10)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,5,7,15)(2,4,9,6)(3,12,13,8)(10,16,14,11)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=de,e^b=eg,f^c=fh,e^d=eg49 conjugacy classes — view the full character table.
| maximal subgroup of | #153623, C2×#20003, #6358398, … |
| maximal normal subgroup of | #153623, C2×#20003, #6358398, … |
| maximal quotient of | #220812, #220824, #225600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12991);