Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499949 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13331 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
585 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 585 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(11,21)(12,26)(13,14)(16,30)(18,34)(19,20)(22,35)(23,38)(24,27)(25,39)(28,29)(31,40)(32,37)(33,36);(1,2)(3,6)(4,5)(7,8)(9,10)(11,22,23,32)(12,18)(13,19)(14,20)(15,17)(16,24,31,25)(21,37,38,35)(26,34)(27,30,39,40)(28,33)(29,36);(2,5)(6,8)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,31)(18,35)(19,32)(20,22)(21,26)(28,39)(29,38)(30,34)(33,37)(36,40);(1,3,4,7)(2,6,5,8)(10,17)(12,28)(16,31)(19,20)(21,38)(22,32)(26,29)(27,39)Polycyclic presentation:
2^9|b^2=e,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239792, #72239816, #72239832, … |
| maximal normal subgroup of | #72239792, #72239816, #72239832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414931);