Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1050 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)(7,12,19,28)(8,20)(9,11)(10,15)(13,22)(14,23)(16,27,18,24)(17,31,33,35)(21,25)(26,30,36,32)(29,34);(1,2)(3,4)(6,11,15,14)(7,16,19,18)(8,21)(13,20,29,25)(17,32,33,30)(22,34)(26,36)(31,35);(2,4)(5,7)(6,12,15,28)(8,17)(9,18)(10,19)(11,24,14,27)(13,26,29,36)(16,23)(20,31,25,35)(21,33)(22,32)(30,34);(5,8,9,22,10,21,23,34)(6,13,14,20,15,29,11,25)(7,17,18,32,19,33,16,30)(12,26,27,31,28,36,24,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281950);