Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
774 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)(6,13,17,27)(7,18,20,28)(8,23)(9,10)(11,24)(14,31)(15,16,25,35)(19,33)(21,29,26,22)(32,36);(5,6,8,13,11,16,23,15,12,17,10,27,24,35,9,25)(7,19,21,36,22,34,18,14,20,33,26,31,29,30,28,32);(2,4)(5,7)(6,14)(8,18)(9,26)(10,28)(11,29)(12,20)(13,30)(15,33)(16,36)(17,32)(19,25)(21,23)(22,24)(27,34)(31,35);(7,20)(14,32)(18,28)(19,33)(21,26)(22,29)(30,34)(31,36)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250649, #72250665, #72250684, … |
| maximal normal subgroup of | #72250649, #72250665, #72250684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406501);