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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1350 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)(10,13)(11,15)(17,18)(19,24)(20,31)(21,25)(22,26)(23,27)(28,29)(30,32);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(17,19,23,21)(18,24,27,25)(20,28,26,32)(22,30,31,29);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,31,18,23,26,22,27)(19,28,29,24,21,32,30,25);(3,8)(6,7)(10,13)(14,16)(17,21,23,19)(18,25,27,24)(20,32,26,28)(22,29,31,30);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,22,23,31)(18,26,27,20)(19,29,21,30)(24,28,25,32)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=fgi,d^2=i,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=ehi,d^c=dh,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, 6956710);