Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884809 |
| 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 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(17,18,20,24,21,25,28,23)(19,22,26,30,27,31,32,29);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,3)(2,7)(4,6)(5,8)(10,13)(11,15)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(2,6)(3,8)(9,11,12,15)(10,14,13,16);(1,4,5,7)(2,3,6,8)(10,13)(14,16)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6844777);