Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18,23,25)(19,26,30,27)(20,22,31,21)(24,28,32,29);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,23,25)(19,27,30,26)(20,22,31,21)(24,29,32,28);(1,3,5,4)(2,7,6,8)(17,19,21,28,23,30,22,29)(18,24,31,27,25,32,20,26);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,25,23,31,21,18)(19,27,29,24,30,26,28,32);(1,4,5,3)(2,7,6,8)(17,21,23,22)(18,20,25,31)(19,28,30,29)(24,26,32,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fgi,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7501407);