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 | |
1193 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)(7,16,9,12)(8,20,22,30)(10,15)(11,14)(13,23,29,25)(17,33)(18,27)(19,32,24,28)(21,36)(26,34)(31,35);(5,7,10,19,11,9,18,24)(6,12,15,28,14,16,27,32)(8,21,23,35,22,17,25,34)(13,26,20,36,29,31,30,33);(5,8,11,22)(6,13,14,29)(7,17,9,21)(10,25,18,23)(12,26,16,31)(15,30,27,20)(19,35,24,34)(28,33,32,36);(2,4)(5,9,10,24,11,7,18,19)(6,12,27,32,14,16,15,28)(8,21,25,34,22,17,23,35)(13,26,20,36,29,31,30,33)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=i,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 343879);