Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18,23,26)(19,20,22,25)(21,24,31,32)(27,28,30,29);(1,3,5,8)(2,6,4,7)(9,11,14,10)(12,15,13,16)(17,19,23,22)(18,20,26,25)(21,27,31,30)(24,28,32,29);(9,12)(10,15)(11,16)(13,14)(17,20,22,26,23,25,19,18)(21,29,30,24,31,28,27,32);(1,4,5,2)(3,6,8,7)(9,13,14,12)(10,16,11,15)(17,21,23,31)(18,24,26,32)(19,27,22,30)(20,28,25,29);(10,11)(15,16)(17,19,23,22)(18,25,26,20)(21,30,31,27)(24,28,32,29)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6950498);