Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884937 |
| 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 | #28488 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,8,5)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,22)(23,31)(24,29)(25,32);(1,3)(2,4)(5,7)(6,8)(10,13)(14,15)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,4,6,5)(2,3,7,8)(9,11,12,16)(10,14,13,15)(18,22)(19,26)(24,29)(27,30);(1,5,6,4)(2,3,7,8)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6871971);