Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425091 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
939 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)(5,6,12,15,11,16,10,9)(7,17,19,13,20,30,25,28)(8,14,23,31,24,29,26,22)(18,32,34,27,21,35,33,36);(5,7,11,20)(6,13,16,28)(8,21)(9,17,15,30)(10,25,12,19)(14,27)(18,24)(22,32)(23,33)(26,34)(29,36)(31,35);(5,8,12,23,11,24,10,26)(6,14,9,22,16,29,15,31)(7,18,25,33,20,21,19,34)(13,27,30,35,28,36,17,32);(2,4)(5,9,12,16,11,15,10,6)(7,13,25,30,20,28,19,17)(8,22,23,29,24,31,26,14)(18,27,33,35,21,36,34,32)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=ef,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 344750);