Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477006 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17101 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,19,8)(3,12,16,23,13,21,15,7)(4,9,17,24,18,27,20,26)(14,28,30,22,29,32,31,25);(1,3,5,16,6,13,19,15)(2,7,10,12,11,23,8,21)(4,14,17,30,18,29,20,31)(9,25,24,28,27,22,26,32);(2,8)(3,13)(4,17)(5,19)(7,21)(10,11)(12,23)(14,30)(18,20)(22,25)(24,26)(29,31);(1,4)(2,9)(3,14)(5,20)(6,18)(7,22)(8,24)(10,26)(11,27)(12,28)(13,29)(15,30)(16,31)(17,19)(21,32)(23,25)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384202);