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 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1475 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,25)(19,24,29,32)(20,21,30,22)(26,28,31,27);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,25,32)(20,26,30,31)(21,28,22,27);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,20)(18,21)(19,26)(22,25)(23,30)(24,28)(27,32)(29,31);(9,11,12,16)(10,14,13,15)(19,27)(20,30)(21,22)(24,26)(28,29)(31,32);(2,7)(6,8)(17,21,23,22)(18,20,25,30)(19,28,29,27)(24,26,32,31)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=g,e^2=i,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,e^c=ei,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, 7455668);