Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37616 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(7,13)(10,14)(15,16)(18,19)(21,22,27,31,26,23,25,29)(24,32,35,28,33,36,34,30);(2,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,19)(16,18)(17,20)(21,23,26,22)(24,28,33,30)(25,29,27,31)(32,34,36,35);(5,7,9,13)(6,10,12,14)(8,16,17,15)(11,19,20,18)(21,24,26,33)(22,28,23,30)(25,35,27,34)(29,36,31,32);(5,8,9,17)(6,11,12,20)(7,15,13,16)(10,18,14,19)(22,23)(24,33)(29,31)(34,35);(6,12)(10,14)(11,20)(18,19)(22,23)(28,30)(29,31)(32,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6439791);