Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| 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 | |
1214 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)(7,17)(8,23)(9,15)(10,14)(11,16)(12,20)(13,26)(18,33)(19,22)(21,29)(24,28)(25,31)(27,30)(32,36)(34,35);(1,2)(3,4)(5,7,10,21,11,22,9,20)(6,12,15,19,16,29,14,17)(8,24,26,35,27,18,25,36)(13,28,23,32,31,33,30,34);(5,8)(6,13)(7,18)(9,26)(10,25)(11,27)(12,28)(14,23)(15,30)(16,31)(17,32)(19,34)(20,36)(21,35)(22,24)(29,33);(1,3,4,2)(5,6)(7,19)(8,13)(9,15)(10,14)(11,16)(12,21)(17,22)(18,34)(20,29)(23,25)(24,32)(26,30)(27,31)(28,35)(33,36)Polycyclic presentation:
2^9|b^2=f,d^2=ef,e^2=i,f^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,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336698);