Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2099 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ 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:
(5,6,12,18,13,20,29,19)(7,21,14,26,25,33,24,34)(8,23,27,35,28,32,36,22)(9,17,30,15,10,31,11,16);(1,2)(3,4)(5,7,12,14,13,25,29,24)(6,8,19,36,20,28,18,27)(9,23,30,35,10,32,11,22)(15,34,17,33,16,26,31,21);(2,4)(5,8,9,21,29,36,11,34,13,28,10,33,12,27,30,26)(6,14,15,23,18,25,31,35,20,24,16,32,19,7,17,22)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=hi,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=fh,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250554, #72250558, #72250570, … |
| maximal normal subgroup of | #72250554, #72250558, #72250570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56331);