Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2131 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)(5,6,11,16,12,17,10,29)(7,18,21,33,22,32,20,13)(8,23,25,36,26,35,30,14)(9,27,24,34,19,31,28,15);(5,7,10,20,12,22,11,21)(6,13,16,18,17,33,29,32)(8,24,25,19,26,28,30,9)(14,34,35,27,36,15,23,31);(2,4)(5,8,12,26)(6,14,17,36)(7,19,22,9)(10,25,11,30)(13,31,33,27)(15,18,34,32)(16,35,29,23)(20,28,21,24);(5,9)(6,15)(7,8)(10,24)(11,28)(12,19)(13,23)(14,18)(16,31)(17,34)(20,25)(21,30)(22,26)(27,29)(32,36)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302827);