Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| 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 | |
925 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)(5,6,12,15,10,9,11,26)(7,13,20,28,18,17,19,31)(8,21,24,30,23,14,25,32)(16,33,22,29,34,27,35,36);(5,7,10,18)(6,13,9,17)(8,22,23,35)(11,20,12,19)(14,29,21,36)(15,31,26,28)(16,25,34,24)(27,32,33,30);(5,8,10,23)(6,14)(7,16,18,34)(9,21)(11,25,12,24)(13,27)(15,32)(17,33)(19,35,20,22)(26,30)(28,36)(29,31);(2,4)(5,9)(6,10)(7,17)(8,14,23,21)(11,26)(12,15)(13,18)(16,27,34,33)(19,31)(20,28)(22,29,35,36)(24,30,25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,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, 370349);