Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425091 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
939 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)(5,6,11,16)(7,17)(8,21)(9,14,22,35)(10,15,27,31)(12,28)(13,23)(18,33)(19,30)(20,36)(24,34,26,29)(25,32);(5,7,11,20)(6,12)(8,22)(9,23)(10,19,27,28)(13,29,21,34)(14,33,35,32)(15,17)(16,30)(18,26)(24,25)(31,36);(5,8,10,25,11,23,27,18)(6,13,31,32,16,21,15,33)(7,9,19,26,20,22,28,24)(12,29,36,14,30,34,17,35);(2,4)(5,9)(6,14)(7,18)(8,19)(10,26)(11,22)(12,21)(13,30)(15,29)(16,35)(17,32)(20,25)(23,28)(24,27)(31,34)(33,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 342676);