Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #42910 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,13,16)(11,12,15,14)(17,18,22,27)(19,29,32,24)(20,26,34,44)(21,28,23,25)(30,42,33,39)(31,40,36,38)(35,41,37,43)(45,47,46,48);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,33)(21,36)(22,32)(23,31)(25,38)(26,42)(27,29)(28,40)(30,34)(35,45)(37,46)(39,44)(41,48)(43,47);(1,3)(2,5)(4,8)(6,7)(9,12,13,14)(10,15,16,11)(18,25)(20,34)(21,23)(24,38)(26,43)(27,28)(29,40)(30,33)(31,36)(39,48)(41,44)(42,47);(3,8)(5,7)(11,15)(12,14)(17,20,22,34)(18,26,27,44)(19,30,32,33)(21,37,23,35)(24,39,29,42)(25,41,28,43)(31,46,36,45)(38,47,40,48);(2,6)(5,7)(10,16)(11,15)(17,21,22,23)(18,25,27,28)(19,31,32,36)(20,35,34,37)(24,40,29,38)(26,43,44,41)(30,45,33,46)(39,47,42,48)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,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, 7143288);