Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #512070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,15)(8,11)(9,21)(10,24)(12,17)(13,23)(14,20)(16,31)(18,29)(19,35)(22,28)(25,32)(26,33)(27,36)(30,34);(1,2,4,3)(9,19)(13,27)(15,31)(17,33)(21,35)(22,34)(23,36)(29,32);(6,11)(9,19)(12,26)(14,24)(17,33)(22,34)(23,36)(28,30);(1,3,4,2)(5,7,10,25,8,16,20,18)(6,12,14,30,11,26,24,28)(9,22,23,33,19,34,36,17)(13,29,21,31,27,32,35,15)Polycyclic presentation:
2^9|b^2=efi,d^2=eh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=ghi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419497);