Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470122 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,17)(15,19)(18,20);(2,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,20)(16,18)(17,19);(7,11)(9,12)(13,14)(15,20)(16,18)(17,19);(7,11)(9,12)(14,18)(17,20);(6,10)(9,12)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282629, #72283195, #72315732, … |
| maximal normal subgroup of | #72282629, #72283195, #72315732, … |
| maximal quotient of | #72310236, #72316265, #72317888 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377318);