Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422233 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,18)(7,19)(8,25)(9,28,30,34)(10,16,11,17)(13,23)(14,26)(15,20,36,29)(21,31)(22,33)(24,32)(27,35);(1,2)(3,4)(5,7,10,23,12,24,11,22)(6,13,16,32,18,33,17,19)(8,9,27,20,21,30,26,29)(14,34,25,15,35,28,31,36);(2,4)(5,8)(6,14)(7,20)(9,23)(10,26)(11,27)(12,21)(13,15)(16,31)(17,25)(18,35)(19,28)(22,30)(24,29)(32,34)(33,36);(5,9,10,20,12,30,11,29)(6,15,16,28,18,36,17,34)(7,21,22,27,24,8,23,26)(13,31,19,35,33,25,32,14)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^c=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, 329429);