Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,30,21,9,15,12,17,31,20,11,7,18,14)(8,22,27,36,32,33,25,35,29,23,26,13,28,34,24,19);(2,4)(5,7,11,6,12,21,30,17)(8,23,32,33,29,22,28,34)(9,15,10,20,18,14,31,16)(13,27,35,25,36,26,19,24);(2,4)(5,8,11,32,12,29,30,28)(6,13,21,35,17,36,7,19)(9,27,10,25,18,26,31,24)(14,33,16,22,15,34,20,23)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321294, #72321310, #72321333, … |
| maximal normal subgroup of | #72321294, #72321310, #72321333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56581);