Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,13,11,15,9,16)(7,17,20,28,21,12,19,26)(8,22,23,14,24,29,25,31)(18,32,33,30,34,36,35,27);(1,2)(3,4)(5,7,11,21)(6,12,15,17)(8,18,24,34)(9,20,10,19)(13,28,16,26)(14,30,31,27)(22,36,29,32)(23,35,25,33);(2,4)(6,13)(9,10)(12,26)(14,29)(15,16)(17,28)(19,20)(22,31)(23,25)(27,32)(30,36)(33,35);(5,8)(6,14)(7,18)(9,25)(10,23)(11,24)(12,27)(13,29)(15,31)(16,22)(17,30)(19,35)(20,33)(21,34)(26,32)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290354);