Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,15)(8,12)(9,13)(10,19)(11,14)(16,18)(17,20);(5,7,10,12,11,17,9,16)(6,8,13,15,14,18,19,20);(1,2,3,4)(5,8)(6,12)(7,13)(9,15)(10,20)(11,18)(14,16)(17,19);(5,8)(6,12)(7,13)(9,15)(10,20)(11,18)(14,16)(17,19);(5,9,11,10)(6,13,14,19)(7,16,17,12)(8,15,18,20)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72285953, #72286003, #72286100, … |
| maximal normal subgroup of | #72285953, #72286003, #72286100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350523);