Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,13)(8,20)(9,23)(10,14)(11,22)(12,15)(16,17)(18,29)(19,31)(21,32)(24,25)(26,28)(27,30)(33,36)(34,35);(1,2,4,3)(5,7)(6,13)(8,21)(9,17)(10,27)(11,18)(12,19)(14,30)(15,31)(16,23)(20,32)(22,29)(24,33)(25,36)(26,35)(28,34);(5,8,20,6,11,23,26,14,12,24,25,15,10,28,9,22)(7,16,32,30,18,33,35,31,19,34,36,29,27,21,17,13);(1,3,4,2)Polycyclic presentation:
2^9|b^2=e,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410320);