Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
695 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)(5,6,11,16,12,15,10,17)(7,18,21,30,22,13,28,34)(8,23,9,14,25,32,26,33)(19,36,20,29,27,31,24,35);(5,7,10,21,12,22,11,28)(6,13,17,34,15,18,16,30)(8,24,26,19,25,20,9,27)(14,31,23,35,33,36,32,29);(5,8,10,26,12,25,11,9)(6,14,17,23,15,33,16,32)(7,19,21,20,22,27,28,24)(13,29,34,31,18,35,30,36);(2,4)(5,9,10,25,12,26,11,8)(6,14,16,23,15,33,17,32)(7,20,21,19,22,24,28,27)(13,29,30,31,18,35,34,36)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,d^2=f,f^2=i,g^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=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386662);