Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394267 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,11)(7,12,8,10)(13,14)(15,17)(16,18)(19,20);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,20)(18,19);(5,8,9,7)(6,10,11,12)(13,14)(15,19,20,17)(16,18);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251263, #72251280, #72251281 |
| maximal normal subgroup of | #72251263, #72251280, #72251281 |
| maximal quotient of | #72255677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305253);