Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 | #24438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1640 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)(11,15,16,13)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(19,22)(20,27)(25,31)(28,32)(29,30);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,21,22)(18,24,23,26)(20,29,30,27)(25,32,28,31);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(20,30)(25,28)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7496996);