Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480482 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1978 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)(5,6,10,14,11,15,26,24)(7,16,19,17,20,9,13,25)(8,12,23,29,21,27,34,35)(18,30,33,31,32,22,28,36);(5,7,17,6,11,20,25,15)(8,18,31,12,21,32,36,27)(9,24,10,13,16,14,26,19)(22,35,23,28,30,29,34,33);(2,4)(5,8)(6,12)(7,18)(9,22)(10,23)(11,21)(13,28)(14,29)(15,27)(16,30)(17,31)(19,33)(20,32)(24,35)(25,36)(26,34);(8,21)(12,27)(18,32)(22,30)(23,34)(28,33)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=egi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401932);