Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,11)(7,15)(8,9)(12,16)(13,14)(17,18);(1,2)(3,4)(5,6,9,13,10,14,8,11)(7,16,15,20,17,12,18,19);(5,7)(6,12)(8,15)(9,18)(10,17)(11,19)(13,20)(14,16);(2,4)(6,13)(8,9)(11,14)(12,20)(15,18)(16,19)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284893, #72285492, #72285600, … |
| maximal normal subgroup of | #72284893, #72285492, #72285600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327968);