Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2788 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)(21,22,26,29,27,24,25,30)(23,31,34,35,32,36,33,28);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,35)(25,34)(26,33)(27,32)(29,36)(30,31);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,22)(23,28,32,35)(25,30,26,29)(31,34,36,33);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,24)(23,32)(29,30)(33,34);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,29,24,30)(23,33,32,34)(28,31,35,36)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=hi,e^2=i,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, 6438232);