Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 6291456 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6322 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)(7,8)(10,12)(14,17)(18,19);(5,6)(7,8)(13,14,16,17)(15,19,20,18);(2,4)(6,8)(13,14)(15,19)(16,17)(18,20);(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)(13,15,16,20)(14,18,17,19)Polycyclic presentation:
2^9|b^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
| maximal subgroup of | #72263450, #72263452, #72263478, … |
| maximal normal subgroup of | #72263450, #72263452, #72263478, … |
| maximal quotient of | #72265189, #72265209, #72332688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213559);