Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1196 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)(7,12)(8,20)(9,22)(10,14)(11,25)(13,21)(15,24)(16,23)(17,29)(18,35)(19,31)(26,27)(28,33)(30,34)(32,36);(5,7,11,18,10,19,24,34)(6,12,15,30,14,31,25,35)(8,16,21,33,22,29,26,36)(9,23,13,32,20,17,27,28);(5,8)(6,13)(7,16)(9,15)(10,22)(11,26)(12,28)(14,27)(17,30)(18,36)(19,29)(20,25)(21,24)(23,35)(31,32)(33,34);(2,4)(5,9,11,27,10,20,24,13)(6,8,15,26,14,22,25,21)(7,17,34,28,19,23,18,32)(12,29,35,36,31,16,30,33)Polycyclic presentation:
2^9|b^2=f,d^2=efi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=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, 336671);