Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7199982 |
| 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 | #36127 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1717 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)(3,6)(4,5)(7,8)(9,10,12,15)(11,13,16,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,31)(29,32);(2,5)(3,7)(17,19)(18,22)(20,29)(21,26)(23,25)(24,32)(27,30)(28,31);(10,13)(11,16)(14,15)(19,26)(22,25)(27,29)(31,32);(1,3,4,7)(2,6,5,8)(18,23)(22,25)(24,28)(31,32);(2,5)(6,8)(17,20,21,30)(18,24,23,28)(19,27,26,29)(22,31,25,32)Polycyclic presentation:
2^9|a^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7139206);