Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×H(3,𝔽4) | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
932 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 932 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,15)(13,25)(14,27)(16,23)(17,21)(18,24)(20,36)(22,35)(26,32)(28,33)(29,39)(30,38)(31,40)(34,37);(1,3,5,4)(2,6,8,7)(10,15)(11,20)(14,24)(16,30)(17,25)(19,34)(22,37)(23,40)(26,28)(29,31)(35,36)(38,39);(1,3,5,4)(2,7,8,6)(9,11,21,39,12,22,13,23)(10,16,25,35,15,29,17,19)(14,20,26,40,24,37,28,38)(18,31,32,36,27,30,33,34);(1,4,5,3)(2,7,8,6)(10,15)(16,29)(17,25)(18,27)(19,35)(30,31)(32,33)(34,36)Polycyclic presentation:
2^9|b^2=ei,c^2=ef,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72271412 |
| maximal normal subgroup of | #72271412 |
| maximal quotient of | #72271458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410670);