Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #30714 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(18,23)(19,26)(21,28)(24,31)(27,29)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,28)(21,25)(22,23)(24,30)(26,31)(29,32);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,28,32,22,30,21,26)(18,24,20,31,23,27,25,29);(3,8)(6,7)(11,16)(13,15)(17,20)(18,21)(19,29)(22,25)(23,28)(24,32)(26,27)(30,31);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,28)(18,20,23,25)(19,26,30,32)(24,31,27,29)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6934779);