Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #502133 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,25,33,24,31,8,14,23,32,11,16,10,13)(7,17,20,34,36,29,21,30,19,12,18,27,22,35,26,28);(2,4)(5,7,9,20,25,36,24,21,8,19,23,18,11,22,10,26)(6,12,15,27,33,35,31,28,14,17,32,34,16,29,13,30);(6,13)(7,18)(9,10)(11,25)(12,17)(14,31)(15,16)(19,20)(21,36)(22,26)(23,24)(27,28)(30,34)(32,33);(2,4)(6,14)(12,17)(13,31)(15,32)(16,33)(27,34)(28,30)(29,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250107 |
| maximal normal subgroup of | #72250107 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409329);