Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1708 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)(9,10)(11,13)(12,14)(15,16)(18,23)(19,21)(24,30)(25,29)(26,28)(31,32);(9,11)(10,13)(12,15)(14,16)(17,18,22,25)(19,23,21,29)(20,24,27,31)(26,30,28,32);(1,3,4,8)(2,5,6,7)(11,15)(13,16)(17,19,22,21)(18,23,25,29)(20,26,27,28)(24,30,31,32);(11,15)(13,16)(17,20,22,27)(18,24,25,31)(19,26,21,28)(23,30,29,32);(10,14)(13,16)(20,27)(24,31)(26,28)(30,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382546);