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 | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26)(19,25,24,23)(20,30,31,29)(22,28,32,27);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,24)(18,23,26,25)(20,28,31,27)(22,29,32,30);(9,11)(10,13)(12,16)(14,15)(17,20,22,25,21,31,32,23)(18,24,27,30,26,19,28,29);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,19)(18,25)(20,28)(21,24)(22,29)(23,26)(27,31)(30,32);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,24)(20,31)(23,25)(29,30)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=fi,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=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, 6937534);