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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1483 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,22,20)(19,23,26,28)(21,24,30,29)(25,31,27,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,26)(18,23,20,28)(21,25,30,27)(24,31,29,32);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,22,18)(19,23,26,28)(21,29,30,24)(25,31,27,32);(10,14)(13,16)(19,25)(21,30)(23,31)(24,29)(26,27)(28,32);(1,4,5,3)(2,7,6,8)(19,26)(23,28)(25,27)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7435491);