Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1239 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)(5,6,11,16,12,18,10,17)(7,19,23,32,24,34,22,13)(8,25,27,14,28,30,15,9)(20,36,26,29,35,21,31,33);(5,7,10,22,12,24,11,23)(6,13,16,19,18,32,17,34)(8,26,27,35,28,31,15,20)(9,29,30,36,14,33,25,21);(5,8,11,27,12,28,10,15)(6,14,17,25,18,9,16,30)(7,20,22,31,24,35,23,26)(13,21,19,33,32,36,34,29);(2,4)(5,9,12,14)(6,15,18,27)(7,21,24,36)(8,16,28,17)(10,25,11,30)(13,31,32,26)(19,35,34,20)(22,33,23,29)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341118);