Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,15,18,11)(8,19)(9,22)(10,13)(12,21)(14,24)(16,30,31,36)(17,26,23,28)(20,33)(25,32,29,35)(27,34);(1,2,3,4)(5,7)(6,11)(8,16)(9,23)(10,18)(12,25)(13,15)(14,28)(17,24)(19,30)(20,31)(21,35)(22,26)(27,29)(32,34)(33,36);(5,8,9,21,10,20,24,34)(6,12,14,19,13,27,22,33)(7,16,23,35,18,31,17,32)(11,25,28,30,15,29,26,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130115, #130117, #130119, … |
| maximal normal subgroup of | #130115, #130117, #130119, … |
| maximal quotient of | #110050, #110052, #110056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3935);