Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,22,36)(20,30,32,38)(21,33,34,26)(25,37,41,28)(29,43,35,39)(31,45,46,44)(40,42,47,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,29,34,35)(25,38,41,30)(26,39,33,43)(31,42,46,48)(40,45,47,44);(9,11)(10,13)(12,16)(14,15)(17,20,23,32)(18,25,27,41)(19,28,22,37)(21,31,34,46)(24,38,36,30)(26,40,33,47)(29,42,35,48)(39,45,43,44);(3,8)(5,7)(11,16)(13,15)(17,21,23,34)(18,26,27,33)(19,29,22,35)(20,31,32,46)(24,39,36,43)(25,40,41,47)(28,42,37,48)(30,44,38,45);(2,6)(5,7)(10,14)(13,15)(18,27)(21,34)(24,36)(25,41)(29,35)(30,38)(31,46)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7141551);