Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,20,12)(8,21)(9,26,22,35)(10,16)(11,29)(13,24)(14,30,31,27)(15,28)(18,33)(19,32,23,36)(25,34);(1,2)(3,4)(5,7,11,19,10,20,28,23)(6,12,15,32,16,17,29,36)(8,22,24,30,25,9,18,27)(13,31,21,35,33,14,34,26);(2,4)(5,8,11,24,10,25,28,18)(6,13,15,21,16,33,29,34)(7,9,23,30,20,22,19,27)(12,31,36,26,17,14,32,35);(5,9)(6,14)(7,18)(8,23)(10,22)(11,30)(12,21)(13,32)(15,35)(16,31)(17,34)(19,25)(20,24)(26,29)(27,28)(33,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,g^c=gi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296517);