Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499757 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61084 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,14)(9,26)(10,28)(11,16)(12,19)(13,22)(15,24)(18,31)(20,36)(21,35)(23,34)(25,33)(27,32)(29,30);(1,2,4,3)(7,18)(8,22)(9,25)(10,24)(12,30)(13,14)(15,28)(17,31)(19,29)(20,27)(21,23)(26,33)(32,36)(34,35);(5,7,11,21)(6,12,16,32)(8,23,25,18)(9,27,22,19)(10,29,24,20)(13,30,26,36)(14,17,33,35)(15,34,28,31);(1,3,4,2)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407910);