Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1720 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)(9,10)(11,14)(12,16)(13,15)(17,18,23,28)(19,24,32,39)(20,25,22,27)(21,26,34,41)(29,36,44,47)(30,37,31,38)(33,40,35,42)(43,46,45,48);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,31)(21,29)(22,30)(23,32)(25,38)(26,36)(27,37)(28,39)(33,45)(34,44)(35,43)(40,48)(41,47)(42,46);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20)(18,25)(21,34)(22,23)(26,41)(27,28)(29,43)(30,31)(36,46)(37,38)(44,45)(47,48);(9,12,13,11)(10,16,15,14)(17,21)(18,26)(19,29)(20,33)(22,35)(23,34)(24,36)(25,40)(27,42)(28,41)(30,43)(31,45)(32,44)(37,46)(38,48)(39,47);(1,4,5,3)(2,6,8,7)(19,30)(20,22)(21,35)(24,37)(25,27)(26,42)(31,32)(33,34)(38,39)(40,41)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6497638);