Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500599 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13316 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
821 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)(7,19)(8,23)(9,13)(10,17)(11,16)(12,18)(14,24)(15,22)(20,31)(21,29)(25,28)(26,36)(27,30)(32,35)(33,34);(1,2)(3,4)(5,7,10,9,12,22,11,21)(6,13,16,19,18,29,17,15)(8,24,26,20,27,34,25,35)(14,23,32,28,33,30,31,36);(2,4)(5,8)(6,14)(7,20)(9,24)(10,25)(11,26)(12,27)(13,28)(15,30)(16,31)(17,32)(18,33)(19,23)(21,34)(22,35)(29,36);(5,9,10,22,12,21,11,7)(6,15,16,13,18,19,17,29)(8,24,25,35,27,34,26,20)(14,30,31,28,33,23,32,36)Polycyclic presentation:
2^9|b^2=h,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407795);