Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,21)(9,17)(10,15)(11,16)(13,23)(14,19)(18,33)(20,29)(22,30)(24,31)(25,32)(26,28)(27,34)(35,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,27)(14,15)(16,29)(17,28)(18,24)(21,30)(23,34)(25,35)(31,33)(32,36);(5,8,9,23,11,24,10,25)(6,13,15,21,16,32,17,31)(7,18,19,34,20,22,26,35)(12,27,28,33,29,36,14,30);(1,3,4,2)(5,7)(6,14)(8,18)(9,19)(10,26)(11,20)(12,15)(13,30)(16,28)(17,29)(21,27)(22,24)(23,34)(25,35)(31,36)(32,33)Polycyclic presentation:
2^9|c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330721);