Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #24250 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 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)(18,22)(23,29)(24,25)(30,31);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,23,26,29)(21,24,28,25)(27,31,32,30);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,26)(21,27)(22,29)(24,31)(25,30)(28,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,24)(19,26)(21,28)(22,25)(23,30)(29,31);(1,4,5,3)(2,7,6,8)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fgi,e^2=fgi,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,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7496975);