Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500596 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13341 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 663 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 36 points:
(5,6)(7,12)(8,21)(9,16)(10,17)(11,20)(13,18)(14,23)(15,27)(19,29)(22,36)(24,33)(25,34)(26,30)(28,32)(31,35);(1,2,3,4)(5,7,12,6,13,17,10,18)(8,22,24,31,26,14,25,19)(9,16,11,15,28,32,27,20)(21,35,34,36,30,29,33,23);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,29)(12,25)(13,26)(15,30)(16,33)(17,31)(18,22)(20,21)(27,35)(28,36)(32,34);(5,9,10,27,13,28,12,11)(6,15,7,16,18,20,17,32)(8,23,24,35,26,36,25,29)(14,30,19,33,22,21,31,34)Polycyclic presentation:
2^9|b^2=g,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265562, #72265587, #72265609, … |
| maximal normal subgroup of | #72265562, #72265587, #72265609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413494);