Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #31399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,24,22,35)(20,30,32,37)(21,26,34,42)(25,36,40,28)(29,38,33,41)(31,44,46,47)(39,45,48,43);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,33,34,29)(25,37,40,30)(26,41,42,38)(31,45,46,43)(39,44,48,47);(11,15)(13,16)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,34,46)(24,37,35,30)(26,39,42,48)(29,43,33,45)(38,47,41,44);(1,3,5,4)(2,6,7,8)(17,21,23,34)(18,26,27,42)(19,29,22,33)(20,31,32,46)(24,38,35,41)(25,39,40,48)(28,43,36,45)(30,44,37,47);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(21,34)(26,42)(29,33)(31,46)(38,41)(39,48)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6656510);