Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | #35189 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18,23,27)(19,24,20,29)(21,26,22,31)(25,28,32,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,30)(22,28)(25,26)(27,29)(31,32);(3,8)(5,7)(10,13)(14,16)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25);(10,13)(14,16)(19,20)(24,29)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678612);