Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15,17,30)(8,12)(9,22,24,13)(10,14)(11,18)(16,31,19,33)(20,35,36,28)(21,29)(23,26)(25,32)(27,34);(1,2,3,4)(13,22)(15,30)(28,35)(31,33);(5,7,10,17)(6,11,14,26)(8,19,21,16)(9,23,24,18)(12,27,29,25)(13,30,22,15)(20,32,36,34)(28,31,35,33);(5,8,10,21)(6,12,14,29)(7,16,17,19)(9,20,24,36)(11,25,26,27)(13,28,22,35)(15,31,30,33)(18,32,23,34)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,f^b=fhi,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256625, #72256628, #72256629, … |
| maximal normal subgroup of | #72256625, #72256628, #72256629, … |
| maximal quotient of | #72263702, #72263718, #72263728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401798);