Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
830 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)(5,6)(7,19)(8,15)(9,16)(10,14)(11,26)(12,18)(13,22)(17,24)(20,29)(21,34)(23,31)(25,33)(27,28)(30,36)(32,35);(5,7,10,20,8,21,11,27,12,23,25,35,24,36,9,22)(6,13,16,30,17,32,33,31,18,28,26,34,15,29,14,19);(2,4)(6,14)(7,20)(8,24)(9,10)(11,25)(13,28)(15,16)(17,26)(18,33)(19,34)(21,22)(23,35)(27,36)(30,31);(5,8,12,24)(6,15,18,17)(7,21,23,36)(9,10,11,25)(13,29,28,32)(14,26,33,16)(19,34,31,30)(20,27,35,22)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,g^b=gi,d^c=di,e^c=eh,f^c=fhi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250284, #72250290, #72250324, … |
| maximal normal subgroup of | #72250284, #72250290, #72250324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406535);