Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884809 |
| 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 | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1931 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,20,24)(19,26,27,22)(21,23,29,25)(28,30,32,31);(2,6)(4,7)(10,13)(11,15)(17,19)(18,22)(20,27)(21,28)(23,31)(24,26)(25,30)(29,32);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(18,23)(19,27)(21,29)(22,30)(24,25)(26,31);(2,6)(3,8)(10,13)(14,16)(18,24)(22,26)(23,25)(30,31);(1,4,5,7)(2,8,6,3)Polycyclic presentation:
2^9|a^2=i,c^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6871187);