Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472701 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16832 |
| 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 | |
760 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)(3,12,15,23)(4,8)(5,9)(6,11)(7,14,24,16)(10,21)(13,25,31,29)(17,22,30,32)(18,27)(19,26)(20,28);(1,3,6,15)(2,7,11,24)(4,17)(5,16,21,14)(8,25)(9,23,10,12)(13,19)(18,31)(20,30)(22,27)(26,32)(28,29);(1,4,5,19,6,20,21,18)(2,8,9,26,11,28,10,27)(3,13,16,30,15,31,14,17)(7,22,23,25,24,32,12,29);(2,9)(4,18)(5,21)(7,23)(10,11)(12,24)(13,30)(14,16)(17,31)(19,20)(22,32)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379897);