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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,12)(11,14,16,15)(17,18,22,27)(19,26,30,25)(20,31,21,32)(23,29,24,28);(2,5)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,25)(20,24)(21,23)(22,30)(26,27)(28,32)(29,31);(9,12,13,10)(11,15,16,14)(17,20,24,19,22,21,23,30)(18,25,29,32,27,26,28,31);(11,16)(14,15)(17,21,23,19,22,20,24,30)(18,26,28,32,27,25,29,31);(1,3,4,8)(2,6,5,7)(18,27)(25,26)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fgh,c^2=h,d^2=g,e^2=fi,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=cg,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, 7511233);