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 | #2433 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,20,23)(19,25,27,22)(21,24,29,32)(26,31,28,30);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,29)(31,32);(10,14)(11,15);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,4,5,3)(2,7,8,6)(18,23)(19,27)(24,32)(26,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7437458);