Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394267 |
| 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 | C2×#2408 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3492 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)(7,11)(8,9)(10,12)(17,19);(6,9)(10,11)(13,14)(15,19,20,17)(16,18);(2,4)(13,15)(14,17)(16,20)(18,19);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^9|b^2=eh,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,g^b=gi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72231639, #72231648, #72235081, … |
| maximal normal subgroup of | #72231639, #72231648, #72235081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323063);