Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 | |
1573 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)(20,31,29,39,33,42,36,25)(21,26,35,40,34,41,30,37)(32,44,43,47,45,48,46,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,28)(20,29,33,36)(21,30,34,35)(25,31,39,42)(26,37,41,40)(32,43,45,46)(38,44,47,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,39)(19,29,22,36)(21,32,34,45)(24,31,28,42)(26,38,41,47)(30,43,35,46)(37,44,40,48);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,34)(24,37)(25,38)(27,41)(28,40)(29,43)(31,44)(33,45)(36,46)(39,47)(42,48);(2,6)(5,7)(10,14)(13,15)(18,27)(21,34)(24,28)(25,39)(30,35)(31,42)(32,45)(43,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7216302);