Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1681 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)(11,15)(14,16)(17,18,22,26)(19,23,21,25)(20,28,29,31)(24,30,32,27);(1,2,5,6)(3,7,4,8)(10,13)(11,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,31,32,28);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,19,22,21)(18,25,26,23)(20,27,29,30)(24,28,32,31);(9,11,12,15)(10,14,13,16)(18,26)(23,25)(24,32)(28,31)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7054677);