Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(8,15)(10,16)(11,18)(13,19);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20);(2,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(1,2)(3,4)(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,17,16,20);(2,4)Polycyclic presentation:
2^9|d^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225623, #72234653, #72234655, … |
| maximal normal subgroup of | #72225623, #72234653, #72234655, … |
| maximal quotient of | #72240574, #72240661, #72240665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480912);