Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5802 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)(6,8)(13,14,16,18)(15,17,19,20);(2,4)(10,12);(9,10,11,12)(13,14)(15,17)(16,18)(19,20);(2,4)(5,6)(7,8)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(13,15,16,19)(14,17,18,20)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72263097, #72263100, #72263118, … |
| maximal normal subgroup of | #72263097, #72263100, #72263118, … |
| maximal quotient of | #72323705, #72329924, #72330035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 682510);