Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,19,20,12)(9,13)(10,14)(11,16,23,29)(17,27)(18,28)(21,32,33,25)(22,26)(24,30,34,36)(31,35);(2,4)(5,7,9,17,10,18,22,31)(6,11,13,24,14,23,26,34)(8,16,21,30,20,29,33,36)(12,15,25,27,19,28,32,35);(6,12)(11,15)(13,25)(14,19)(23,28)(24,27)(26,32)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,f^b=fh,f^c=fh,f^d=fi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317378, #72317382, #72317386, … |
| maximal normal subgroup of | #72317378, #72317382, #72317386, … |
| maximal quotient of | C2×#146665 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53861);