Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle, prime radical | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(10,14)(13,15)(17,19,20,27)(18,22,23,25)(21,29,28,26)(24,32,31,30);(1,4,5,3)(2,6,7,8)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,d^2=efh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #99705, #150350, #150694, … |
| maximal normal subgroup of | #99705, #150350, #150694, … |
| maximal quotient of | #244442, #244454, #244471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16808);