Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | #42796 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 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)(17,18,22,28,23,27,19,24)(20,31,29,40,33,42,37,25)(21,34,36,38,35,26,30,41)(32,45,43,39,46,44,48,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,28)(20,29,33,37)(21,30,35,36)(25,31,40,42)(26,38,34,41)(32,43,46,48)(39,44,47,45);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,40)(19,29,22,37)(21,32,35,46)(24,31,28,42)(26,39,34,47)(30,43,36,48)(38,44,41,45);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,30)(20,32)(22,36)(23,35)(24,38)(25,39)(27,34)(28,41)(29,43)(31,44)(33,46)(37,48)(40,47)(42,45);(2,6)(5,7)(10,14)(13,15)(18,27)(21,35)(24,28)(25,40)(30,36)(31,42)(32,46)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7216322);