Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18,23,28)(19,29,32,25)(20,24,34,40)(21,26,36,43)(22,38,30,27)(31,45,35,41)(33,39,47,42)(37,48,46,44);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,30,34,23,32,22,20)(18,24,38,25,28,40,27,29)(21,35,46,33,36,31,37,47)(26,42,48,45,43,39,44,41);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,32,23,34,30,19)(18,25,27,24,28,29,38,40)(21,33,37,35,36,47,46,31)(26,41,44,39,43,45,48,42);(11,16)(13,15)(17,21,23,36)(18,26,28,43)(19,31,32,35)(20,33,34,47)(22,37,30,46)(24,39,40,42)(25,41,29,45)(27,44,38,48);(10,14)(13,15)(21,36)(26,43)(31,35)(33,47)(37,46)(39,42)(41,45)(44,48)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413764);