Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12721 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,14,16,11)(8,17,18,12)(9,13)(10,15)(19,20);(1,2,3,4)(5,7,9,16)(6,10)(8,15,18,20)(11,17)(12,14)(13,19);(6,11)(8,18)(10,17)(12,19)(13,14)(15,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,f^c=fh,f^d=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225792, #72304194 |
| maximal normal subgroup of | #72225792, #72304194 |
| maximal quotient of | #72225790, #72303214 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53876);