Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456417 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1182 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)(5,6,11,15,12,9,10,16)(7,17,20,13,22,30,21,19)(8,23,26,28,27,32,25,14)(18,34,24,31,36,29,35,33);(5,7,11,20,12,22,10,21)(6,13,15,30,9,19,16,17)(8,24,26,36,27,35,25,18)(14,31,23,29,28,33,32,34);(5,8,12,27)(6,14,9,28)(7,18,22,36)(10,26,11,25)(13,29,19,34)(15,32,16,23)(17,33,30,31)(20,35,21,24);(2,4)(5,9)(6,12)(7,19)(8,23)(10,15)(11,16)(13,22)(14,25)(17,20)(18,31)(21,30)(24,29)(26,28)(27,32)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,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,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363614);