Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 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)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,21)(19,27,29,30)(20,28,24,31)(22,25,32,26);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,21,20)(22,31,32,28)(25,27,26,30);(11,15)(13,16)(17,20)(18,19)(21,29)(22,27)(23,24)(25,28)(26,31)(30,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,18)(19,20,29,24)(22,25,32,26)(27,28,30,31);(17,22,23,32)(18,25,21,26)(19,28,29,31)(20,27,24,30)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341564);