Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,22,25,21,19)(20,28,31,24,30,26,29,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,25,22,27,23,18)(20,24,29,28,30,32,31,26);(3,7)(6,8)(10,14)(13,16)(17,20,22,30)(18,24,25,32)(19,26,27,28)(21,31,23,29);(1,3,4,7)(2,6,5,8)(18,25)(19,27)(24,32)(26,28)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #102692, #102694, #102705, … |
| maximal normal subgroup of | #102692, #102694, #102705, … |
| maximal quotient of | #380171, #380219, #380670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23203);