Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1528 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1528 conjugacy classes of subgroups — too many to draw. Only its 130 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)(9,10,15,19)(11,20,23,32)(12,17,13,25)(14,28,29,18)(16,26,33,22)(21,37,30,40)(24,36,27,34)(31,39,35,38);(1,3,4,8)(2,5,6,7)(11,21)(14,29)(16,31)(18,28)(20,37)(22,38)(23,30)(24,27)(26,39)(32,40)(33,35)(34,36);(9,11)(10,16)(12,22)(13,26)(14,30)(15,23)(17,32)(18,35)(19,33)(20,25)(21,29)(24,39)(27,38)(28,31)(34,37)(36,40);(2,6)(5,7)(9,12,15,13)(10,17,19,25)(11,22,23,26)(14,24,29,27)(16,32,33,20)(18,34,28,36)(21,38,30,39)(31,40,35,37)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=i,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,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227698, #72228033, #72228034, … |
| maximal normal subgroup of | #72227698, #72228033, #72228034, … |
| maximal quotient of | #72268020, #72268116, #72268328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407220);