Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39382 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,18,23,27)(19,28,32,43)(20,33,34,25)(21,26,22,37)(24,30,39,31)(29,44,45,42)(35,41,36,40)(38,46,48,47);(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,43)(20,29,35,46,34,45,36,47)(25,38,41,42,33,48,40,44);(2,6)(7,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,36)(22,35)(23,34)(24,38)(26,41)(27,33)(28,42)(30,47)(31,46)(32,45)(37,40)(39,48)(43,44);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(20,34)(25,33)(29,45)(35,36)(38,48)(40,41)(42,44)(46,47);(1,4,5,3)(2,8,6,7)(11,16)(13,14)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,43)(25,40,33,41)(29,46,45,47)(38,44,48,42)Polycyclic presentation:
2^9|a^2=g,b^2=fh,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6523119);