Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518904 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,16)(13,15)(17,18,23,27)(19,22,26,32)(20,25,30,31)(21,28,29,24);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,16,13,15)(17,19)(18,22)(20,28)(21,31)(23,26)(24,30)(25,29)(27,32);(10,11)(15,16)(17,20,22,29,23,30,32,21)(18,24,19,25,27,28,26,31);(9,12,14,13)(10,15,11,16)(17,21,32,30,23,29,22,20)(18,25,26,24,27,31,19,28);(1,4,5,3)(2,7,8,6)(9,13,14,12)(10,16,11,15)(17,22,23,32)(18,26,27,19)(20,29,30,21)(24,31,28,25)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=h,d^2=gi,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=cg,d^b=dh,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511764);