Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,14,11,16,10,15)(7,17,22,34,21,33,20,13)(8,23,25,29,27,9,26,30)(18,36,35,19,32,28,24,31);(1,2)(3,4)(5,7,10,20,11,21,12,22)(6,13,14,17,16,34,15,33)(8,24,25,18,27,35,26,32)(9,28,29,19,23,36,30,31);(2,4)(5,8,12,26,11,27,10,25)(6,9,15,30,16,23,14,29)(7,18,22,24,21,32,20,35)(13,31,33,36,34,19,17,28);(5,9,11,23)(6,8,16,27)(7,19,21,31)(10,30,12,29)(13,32,34,18)(14,26,15,25)(17,35,33,24)(20,28,22,36)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=ei,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383808);