Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884873 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30112 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18,20,29,21,23,22,25)(19,26,30,24,27,32,28,31);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(18,23)(19,27)(25,29)(28,30);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,27)(22,30)(23,31)(25,32)(26,29);(2,6)(5,7)(10,13)(14,16)(18,23)(24,31)(25,29)(26,32);(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,25,23,29)(19,28,27,30)(24,32,31,26)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6873774);