Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
331 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 331 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:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,23,28,22,27)(19,24,29,25,20,31,30,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,29,23,20,22,30)(18,24,26,25,28,31,27,32);(10,14)(13,16)(17,20,23,19)(18,25,28,32)(21,29,22,30)(24,27,31,26);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=egh,b^2=ef,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #311038, #311046, #311054, … |
| maximal normal subgroup of | #311038, #311046, #311054, … |
| maximal quotient of | #360451, #363084, #366414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20804);