Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| 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 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,20,14,21,13,31)(7,22,24,32,25,16,26,36)(8,15,23,33,27,35,28,34)(9,19,11,30,29,18,10,17);(1,2)(3,4)(5,7)(6,15,21,35)(8,11)(9,28)(10,27)(12,24)(13,26)(14,25)(16,18,22,19)(17,32,30,36)(20,33,31,34)(23,29);(2,4)(5,8,9,26,12,28,11,25,14,27,29,24,13,23,10,7)(6,16,17,34,31,36,18,15,21,22,30,33,20,32,19,35)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250815, #72250827, #72250838, … |
| maximal normal subgroup of | #72250815, #72250827, #72250838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56329);