Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45417 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,12,13)(11,14,16,15)(17,18)(19,26)(20,23)(21,24)(22,29)(25,30)(27,31)(28,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,24)(19,22,28,31)(20,25,30,23)(26,27,32,29);(1,3,5,4)(2,7,6,8)(10,13)(11,16)(17,19,20,29,21,28,30,27)(18,22,25,26,24,31,23,32);(1,4,5,3)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,18)(19,22)(20,25)(21,24)(23,30)(26,29)(27,32)(28,31);(10,13)(14,15)(17,20,21,30)(18,23,24,25)(19,27,28,29)(22,26,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6951078);