Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25762 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(19,26)(22,31)(27,32)(28,30);(1,3,4,8)(2,5,6,7)(17,18,20,24)(19,27,26,32)(21,23,29,25)(22,30,31,28);(10,13)(14,16)(17,19,20,26)(18,22,24,31)(21,30,29,28)(23,27,25,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16);(3,8)(5,7)(18,23)(21,29)(22,27)(24,25)(28,30)(31,32)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6602990);