Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470221 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16679 |
| 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 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,15)(3,14,25,4,17,19,10,9)(5,20,24,11,23,29,22,30)(16,26,32,31,28,18,27,21);(1,3,6,10,8,17,7,25)(2,9,12,14,13,4,15,19)(5,16,22,27,23,28,24,32)(11,26,29,31,30,18,20,21);(1,4,8,9)(2,10)(3,15)(5,21)(6,19,7,14)(11,28,30,16)(12,17)(13,25)(18,24)(20,27,29,32)(22,26)(23,31);(1,5,7,24,8,23,6,22)(2,11,12,29,13,30,15,20)(3,16,25,32,17,28,10,27)(4,18,19,21,9,26,14,31)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=efi,d^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,g^c=gi,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377418);