Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 | #24662 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1581 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)(3,6)(4,5)(7,8)(17,18,23,20)(19,26,28,32)(21,25,22,30)(24,31,29,27);(9,10)(11,15)(12,13)(14,16)(17,19,22,27,23,28,21,31)(18,24,25,26,20,29,30,32);(2,5)(6,8)(10,13)(14,15)(17,20,23,18)(19,24,28,29)(21,25,22,30)(26,31,32,27);(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,25,20,30)(19,27,28,31)(24,32,29,26);(1,3,4,7)(2,6,5,8)(19,28)(24,29)(26,32)(27,31)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371899);