Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246079 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1595 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,19)(18,20);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,19)(14,16,20,17);(9,12)(14,15)(16,17)(19,20);(1,3,4,2)(13,16,18,17)(14,15,20,19)Polycyclic presentation:
2^9|b^2=eh,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72309263, #72309271, #72309276, … |
| maximal normal subgroup of | #72309263, #72309271, #72309276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 168552);