Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2240 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)(5,6,10,14,11,16,9,15)(7,17,21,29,22,18,20,12)(8,23,26,31,27,32,25,13)(19,28,34,30,35,36,24,33);(5,7,9,20,11,22,10,21)(6,12,14,17,16,29,15,18)(8,24,25,35,27,34,26,19)(13,30,23,36,31,33,32,28);(2,4)(5,6)(7,18)(8,13)(9,14)(10,15)(11,16)(12,20)(17,22)(19,30)(21,29)(23,25)(24,36)(26,32)(27,31)(28,34)(33,35);(5,8,11,27)(6,13,16,31)(7,19,22,35)(9,26,10,25)(12,28,29,36)(14,32,15,23)(17,33,18,30)(20,34,21,24)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^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,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326209);