Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,30)(21,24)(22,32)(23,28)(26,29)(27,31);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,21,23,29,31,20)(18,24,27,25,28,30,32,26);(11,15)(13,16)(17,20,22,19,23,21,31,29)(18,25,27,30,28,26,32,24);(1,3,4,8)(2,6,5,7)(17,21,22,29,23,20,31,19)(18,26,27,24,28,25,32,30)Polycyclic presentation:
2^8|a^2=efg,b^2=f,c^2=f,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #504996, #505072, #505482, … |
| maximal normal subgroup of | #504996, #505072, #505482, … |
| maximal quotient of | #360932, #362681, #367312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21008);