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 | #44929 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,20,23,24)(21,25,31,32)(27,28,29,30);(1,3)(2,6)(4,7)(5,8)(10,14)(11,12)(17,19,22,23)(18,24,26,20)(21,29,31,27)(25,28,32,30);(9,10)(11,16)(12,15)(13,14)(17,20,23,26,22,24,19,18)(21,30,29,25,31,28,27,32);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,21,22,31)(18,25,26,32)(19,27,23,29)(20,28,24,30);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,14,16)(21,31)(25,32)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6957792);