Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467936 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16595 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
737 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)(3,6)(4,13)(5,14,22,28)(7,11)(8,12)(9,16)(10,21,27,20)(15,23,31,32)(17,25)(18,24)(19,29,30,26);(1,3,7,12,8,11,6,2)(4,9,17,24,18,25,16,13)(5,14,20,27,22,28,21,10)(15,23,19,26,31,32,30,29);(1,4,7,16,8,18,6,17)(2,9,3,13,12,25,11,24)(5,19,20,15,22,30,21,31)(10,26,14,23,27,29,28,32);(1,5,7,20,8,22,6,21)(2,10,3,14,12,27,11,28)(4,15,16,30,18,31,17,19)(9,23,13,29,25,32,24,26)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375132);