Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #37839 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(20,22)(24,31)(27,29)(30,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,24,27,30)(20,28,22,25)(26,32,29,31);(11,15)(13,16)(17,19,20,29,21,27,22,26)(18,24,28,31,23,30,25,32);(2,6)(5,7)(17,20,21,22)(18,25,23,28)(19,26,27,29)(24,31,30,32);(3,8)(5,7)(11,15)(13,16)(18,23)(24,30)(25,28)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6657249);