Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19,23,18)(20,26,30,24)(21,25,22,27)(28,29,32,31);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,32)(27,28);(1,4,5,3)(2,7,8,6)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7438676);