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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1647 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)(11,15,16,13)(17,18,23,26)(19,24,30,32)(20,21,25,22)(27,29,31,28);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,26,32)(20,27,25,31)(21,29,22,28);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,21)(19,27)(22,26)(23,25)(24,29)(28,32)(30,31);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(19,28)(20,25)(21,22)(24,27)(29,30)(31,32);(2,6)(7,8)(10,14)(13,15)(17,21,23,22)(18,25,26,20)(19,29,30,28)(24,31,32,27)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fg,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,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, 7474207);