Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
6738 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)(9,10)(11,12)(13,14)(15,19)(16,18)(17,20);(5,6,7,8)(13,14,16,18)(15,17,20,19);(9,10,11,12)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20);(13,15)(14,17)(16,20)(18,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72260558, #72260742, #72260834, … |
| maximal normal subgroup of | #72260558, #72260742, #72260834, … |
| maximal quotient of | #72320310, C2×#4928968, #72330430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836164);