Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13490 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2073 |
| 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 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,16,12,17,19,15)(7,18,23,33,24,36,22,25)(8,21,26,20,28,34,27,31)(9,29,10,13,30,14,32,35);(1,2)(3,4)(5,7,19,23,12,24,11,22)(6,8,15,26,17,28,16,27)(9,21,32,20,30,34,10,31)(13,33,29,36,35,25,14,18);(2,4)(5,8,12,28)(6,7)(9,25,30,33)(10,18,32,36)(11,26,19,27)(13,34)(14,31)(15,22)(16,23)(17,24)(20,29)(21,35)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322188, #72322199, #72322212, … |
| maximal normal subgroup of | #72322188, #72322199, #72322212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56540);