Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(17,18,22,26)(19,27,20,24)(21,23,30,25)(28,32,29,31);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,16)(18,23)(19,20)(21,30)(24,32)(25,26)(27,31);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,19,22,20)(18,24,26,27)(21,28,30,29)(23,31,25,32);(9,11,12,16)(10,14,13,15)(17,20,22,19)(18,24,26,27)(21,29,30,28)(23,31,25,32);(19,20)(24,27)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371181);