Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425083 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2576 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16117 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13,7,12,6,11)(3,14,4,9,17,25,16,24)(5,19,20,10,21,27,22,28)(15,30,18,29,31,23,32,26);(1,3,6,4,7,17,8,16)(2,9,11,25,12,24,13,14)(5,18,22,31,21,32,20,15)(10,26,19,30,28,29,27,23);(1,4,6,17,7,16,8,3)(2,9,13,14,12,24,11,25)(5,18,22,31,21,32,20,15)(10,23,27,29,28,30,19,26);(1,5)(2,10)(3,15)(4,18)(6,22)(7,21)(8,20)(9,23)(11,19)(12,28)(13,27)(14,29)(16,32)(17,31)(24,30)(25,26)Polycyclic presentation:
2^9|a^2=eg,b^2=fh,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 360994);