Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456670 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,15,10,16,11,9)(7,13,20,18,19,30,21,29)(8,22,25,14,26,24,27,32)(17,33,23,36,35,31,34,28);(5,7)(6,13)(8,23)(9,18)(10,19)(11,20)(12,21)(14,31)(15,29)(16,30)(17,25)(22,28)(24,36)(26,34)(27,35)(32,33);(5,8,12,25,10,26,11,27)(6,14,15,24,16,32,9,22)(7,17,21,34,19,35,20,23)(13,28,29,31,30,36,18,33);(2,4)(5,9,12,16,10,15,11,6)(7,18,21,30,19,29,20,13)(8,24,27,32,26,22,25,14)(17,31,23,36,35,33,34,28)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,d^2=fi,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,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=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370354);