Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,13,15,16)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(10,14)(13,16)(17,19,20,27)(18,22,23,25)(21,29,28,26)(24,32,31,30);(2,6)(3,8)(10,14)(13,16)(18,23)(22,25)(24,31)(30,32)Polycyclic presentation:
2^8|a^2=ef,b^2=efh,c^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142745, #142746, #142777, … |
| maximal normal subgroup of | #142745, #142746, #142777, … |
| maximal quotient of | #383094, #383920, #384237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24657);