Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,12,15,11,9)(7,13,21,18,19,30,20,29)(8,22,27,32,25,24,26,14)(17,33,34,28,35,31,23,36);(1,2)(3,4)(5,7,10,20,12,19,11,21)(6,13,9,18,15,30,16,29)(8,23,26,17,25,34,27,35)(14,31,22,28,32,33,24,36);(2,4)(5,8)(6,14)(7,17)(9,24)(10,27)(11,26)(12,25)(13,28)(15,32)(16,22)(18,31)(19,35)(20,23)(21,34)(29,33)(30,36);(5,9,11,15,12,16,10,6)(7,18,20,13,19,29,21,30)(8,24,26,32,25,22,27,14)(17,31,23,28,35,33,34,36)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=efi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337645);