Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1524 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)(11,13)(12,14)(15,16)(17,18)(19,20)(21,25)(22,27)(23,26)(24,28)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,22)(23,29)(24,30)(25,27)(26,32)(28,31);(9,11)(10,13)(12,16)(14,15)(17,20,23,25,24,31,22,32)(18,21,27,19,28,29,26,30);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,24,29)(18,25,28,32)(19,22,30,23)(20,27,31,26);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,24,23)(18,26,28,27)(19,29,30,21)(20,25,31,32)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fi,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6937134);