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 | #15123 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(6,8)(13,14)(15,16)(17,19)(18,20);(2,3)(10,12)(13,15,17,20)(14,16,19,18);(9,10,11,12)(13,15)(14,18)(16,19)(17,20);(2,3)(5,6)(7,8)(13,16,17,18)(14,15,19,20);(1,3,4,2)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,d^2=g,e^2=f,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 | #72261806, #72261833, #72280497, … |
| maximal normal subgroup of | #72261806, #72261833, #72280497, … |
| maximal quotient of | #72298522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 682439);