Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(17,20);(1,2,3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14)(15,17)(16,18)(19,20);(5,7,8,11)(6,10,9,12)(13,15)(14,17)(16,19)(18,20)Polycyclic presentation:
2^9|b^2=e,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251149, #72251247, #72251281 |
| maximal normal subgroup of | #72251149, #72251247, #72251281 |
| maximal quotient of | #72251472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311225);