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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1345 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)(10,14)(12,15)(17,18)(19,25)(20,22)(21,26)(23,31)(24,27)(28,32)(29,30);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,12,16,15)(17,19)(18,25)(20,26)(21,22)(23,28)(24,29)(27,30)(31,32);(9,11)(10,15)(12,14)(13,16)(17,20,22,18,24,31,23,27)(19,26,21,25,29,32,28,30);(3,8)(6,7)(10,14)(11,16)(17,21,24,28)(18,26,27,32)(19,22,29,23)(20,30,31,25);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,11,14,16)(17,22,24,23)(18,20,27,31)(19,28,29,21)(25,32,30,26)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=gi,d^2=i,e^2=fi,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=dhi,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, 6954910);