Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| 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 | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
449 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 449 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:
(2,4)(5,6)(7,11,16,23)(8,12)(9,20)(10,13)(14,21)(15,27,28,22)(17,31,30,24)(18,26)(19,33)(25,32)(29,34,36,35);(5,7,9,17,10,16,21,30)(6,11,14,24,13,23,20,31)(8,15,19,29,18,28,32,36)(12,22,25,34,26,27,33,35);(1,2,3,4)(5,8,10,18)(6,12,13,26)(7,15,16,28)(9,19,21,32)(11,22,23,27)(14,25,20,33)(17,29,30,36)(24,34,31,35)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=efh,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #94171, #94175, #130058, … |
| maximal normal subgroup of | #94171, #94175, #130058, … |
| maximal quotient of | #110088, #110099, #110102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3928);