Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1429 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)(3,7)(4,6)(5,8)(18,23)(19,26)(21,29)(24,31)(25,30)(27,32);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,23,22,25,29,30)(19,24,26,28,27,31,32,20);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,19,21,26,22,27,29,32)(18,24,23,28,25,31,30,20);(9,11,12,16)(10,14,13,15)(17,20,22,28)(18,19,25,27)(21,24,29,31)(23,26,30,32);(1,4,5,3)(2,6,8,7)(18,25)(20,28)(23,30)(24,31)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=hi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372555);