Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(10,14)(11,12)(17,18,23,20)(19,24,29,30)(21,25,26,22)(27,32,31,28);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,23,29)(18,24,20,30)(21,31,26,27)(22,32,25,28);(9,11,13,12)(10,15,14,16)(17,20,22,21,23,18,25,26)(19,27,28,24,29,31,32,30);(9,11,13,12)(10,16,14,15)(17,21,22,18,23,26,25,20)(19,24,28,31,29,30,32,27);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=g,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7500271);