Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12)(7,13)(8,20,21,30)(10,16,11,15)(14,22,26,23)(17,27)(18,32)(19,25)(24,29)(28,34)(31,35)(33,36);(1,2)(3,4)(6,12)(7,17,19,24)(8,21)(13,27,25,29)(15,16)(18,31,33,34)(22,23)(28,32,35,36);(2,4)(5,7,9,19)(6,13)(8,18,21,33)(10,24,11,17)(12,25)(14,28)(15,27)(16,29)(20,32)(22,31,23,34)(26,35)(30,36);(5,8,9,21)(6,14,12,26)(7,18,19,33)(10,23,11,22)(13,28,25,35)(15,20,16,30)(17,31,24,34)(27,32,29,36)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,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, 330624);