Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,27)(22,28)(23,26)(24,25)(29,32)(30,31);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,25,23,30,22,29)(18,24,28,20,26,32,27,31);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,21,32,23,31,22,24)(18,25,28,30,26,29,27,19);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(18,26)(20,31)(24,32)(27,28);(17,21,23,22)(18,27,26,28)(19,29,30,25)(20,32,31,24)Polycyclic presentation:
2^9|b^2=f,c^2=f,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,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341481);