Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 373 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,22,27,30)(20,24,28,32)(25,29,26,31);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(10,14)(13,15)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^8|a^2=efh,b^2=ef,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125676, #125978, #126445, … |
| maximal normal subgroup of | #125676, #125978, #126445, … |
| maximal quotient of | #385074, #385493, #386782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25650);