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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
830 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)(7,18,22,31)(8,24)(9,15)(10,16)(11,17)(12,21,32,23)(13,27)(14,26)(19,33,25,29)(20,30,36,35)(28,34);(5,7,9,21,11,22,10,23)(6,12,16,31,17,32,15,18)(8,25,27,20,28,19,26,36)(13,33,24,30,14,29,34,35);(5,8)(6,13)(7,19)(9,26)(10,27)(11,28)(12,29)(14,17)(15,24)(16,34)(18,35)(20,21)(22,25)(23,36)(30,31)(32,33);(2,4)(5,8)(6,14)(7,20,22,36)(9,26)(10,27)(11,28)(12,30,32,35)(13,17)(15,34)(16,24)(18,29,31,33)(19,21,25,23)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356250);