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 |
| prime radical | #24281 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
790 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)(5,6,12,27,25,33,10,14,8,15,9,16,11,17,24,32)(7,18,19,28,36,30,26,29,20,13,21,34,23,35,22,31);(2,4)(5,7,9,21,25,36,24,22,8,20,12,19,11,23,10,26)(6,13,16,28,33,35,32,29,15,18,27,34,17,30,14,31);(6,14)(7,19)(9,10)(11,25)(12,24)(13,18)(15,32)(16,17)(20,21)(22,36)(23,26)(27,33)(28,29)(31,34);(2,4)(6,15)(13,18)(14,32)(16,27)(17,33)(28,34)(29,31)(30,35)Polycyclic presentation:
2^9|a^2=fi,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, 409333);