Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,30,38)(20,24,33,41)(21,25,34,42)(27,35,29,37)(28,36,44,47)(31,39,32,40)(43,46,45,48);(2,6)(7,8)(17,19,22,30)(18,23,26,38)(20,28,33,44)(21,27,34,29)(24,36,41,47)(25,35,42,37)(31,45,32,43)(39,48,40,46);(9,11,12,16)(10,13,14,15)(19,27)(20,31)(21,34)(23,35)(24,39)(25,42)(28,44)(29,30)(32,33)(36,47)(37,38)(40,41);(1,3,5,4)(2,7,6,8)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,40)(26,41)(27,43)(29,45)(30,44)(31,34)(35,46)(37,48)(38,47)(39,42);(1,4,5,3)(2,8,6,7)(19,27)(20,32)(21,34)(23,35)(24,40)(25,42)(29,30)(31,33)(37,38)(39,41)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493652);