Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12725 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,13)(8,17,18,11)(9,12)(10,14,20,15)(16,19);(1,2,3,4)(5,7)(6,10,12,20)(8,15)(9,16)(11,19,17,13)(14,18);(6,11)(10,19)(12,17)(13,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=di,f^b=fh,f^c=fh,f^d=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219304, #72219306, #72225886, … |
| maximal normal subgroup of | #72219304, #72219306, #72225886, … |
| maximal quotient of | #72232753, #72232820, #72233308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53888);