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×#3019 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13,14,20)(8,16,18,11)(9,12)(10,15)(17,19);(1,2,3,4)(11,16)(13,20);(5,7)(6,10)(8,17)(9,14)(11,20)(12,19)(13,16)(15,18);(7,14)(10,19)(13,20)(15,17)Polycyclic presentation:
2^9|a^2=g,b^2=eg,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 | #72213329, #72235270, #72235423, … |
| maximal normal subgroup of | #72213329, #72235270, #72235423, … |
| maximal quotient of | #72210479, #72210503, #72231828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401796);