Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,15,28,32)(7,17,20,34,21,12,27,30)(8,22,25,13,24,26,14,9)(18,35,29,31,23,36,33,19);(2,4)(5,7,11,21)(6,12,15,17)(8,23,24,18)(9,19,13,31)(10,27,28,20)(14,33,25,29)(16,34,32,30)(22,36,26,35);(1,2,3,4)(5,8,10,14,11,24,28,25)(6,13,32,26,15,9,16,22)(7,18,20,33,21,23,27,29)(12,19,34,35,17,31,30,36);(5,9)(6,14)(7,19)(8,16)(10,22)(11,13)(12,29)(15,25)(17,33)(18,34)(20,35)(21,31)(23,30)(24,32)(26,28)(27,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293569);