Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,26,11,16,10,15)(7,17,19,12,21,34,20,30)(8,22,24,32,25,13,27,14)(18,31,23,28,35,29,36,33);(1,2,3,4)(5,7,9,19,11,21,10,20)(6,12,15,17,16,30,26,34)(8,23,24,35,25,36,27,18)(13,31,32,33,22,29,14,28);(5,8)(6,13)(7,18)(9,27)(10,24)(11,25)(12,28)(14,15)(16,22)(17,29)(19,36)(20,23)(21,35)(26,32)(30,33)(31,34);(2,4)(5,8)(6,14,16,32)(7,18)(9,27)(10,24)(11,25)(12,29,30,31)(13,15,22,26)(17,33,34,28)(19,36)(20,23)(21,35)Polycyclic presentation:
2^9|a^2=fg,b^2=e,d^2=f,e^2=i,g^2=i,h^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^c=ei,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343356);