Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1828 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)(5,6,10,25,11,9,23,14)(7,15,17,12,18,31,35,24)(8,19,21,28,22,13,26,29)(16,32,33,27,20,30,34,36);(5,7,10,17,11,18,23,35)(6,12,14,15,9,24,25,31)(8,20,21,34,22,16,26,33)(13,27,28,32,19,36,29,30);(5,8,11,22)(6,13,9,19)(7,16,18,20)(10,26,23,21)(12,27,24,36)(14,29,25,28)(15,30,31,32)(17,34,35,33);(2,4)(5,9)(6,11)(7,15)(8,19)(10,14)(12,17)(13,22)(16,30)(18,31)(20,32)(21,28)(23,25)(24,35)(26,29)(27,34)(33,36)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326402);