Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,39,23,29,24,28)(19,30,33,42,34,25,35,46)(20,27,38,26,21,37,36,44)(31,45,48,43,32,41,47,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,34)(18,25,29,30)(20,32,21,31)(22,33,24,35)(26,43,44,40)(27,45,37,41)(28,42,39,46)(36,48,38,47);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,36)(24,38)(25,40)(27,39)(28,37)(29,44)(30,43)(32,34)(33,47)(35,48)(41,46)(42,45);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,27,29,37)(19,32,34,31)(22,36,24,38)(25,41,30,45)(26,28,44,39)(33,47,35,48)(40,42,43,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,33,34,35)(20,36,21,38)(25,42,30,46)(26,37,44,27)(31,47,32,48)(40,45,43,41)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6924543);