Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501810 |
| 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 | C22×D4 | SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(7,16)(9,20)(13,30)(17,33)(19,25)(21,36)(23,26)(31,35);(2,4)(5,6,11,14,8,15,10,24)(7,17,25,35,16,33,19,31)(9,21,26,30,20,36,23,13)(12,27,28,18,29,22,32,34);(5,7,8,16)(6,12,15,29)(9,22,20,27)(10,25,11,19)(13,31,30,35)(14,32,24,28)(17,21,33,36)(18,23,34,26);(2,4)Polycyclic presentation:
2^9|b^2=fh,c^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72322902 |
| maximal normal subgroup of | #72322902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409008);