Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469983 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1426 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)(3,6)(4,5)(7,8)(11,12)(14,16)(17,18,22,27,23,28,21,26)(19,29,31,38,32,42,36,24)(20,25,35,41,34,40,33,39)(30,43,44,47,45,48,46,37);(9,10)(11,16)(12,14)(13,15)(17,19,23,32)(18,24,28,38)(20,30,34,45)(21,31,22,36)(25,37,40,47)(26,29,27,42)(33,44,35,46)(39,43,41,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,30)(21,33)(22,35)(23,34)(24,37)(26,39)(27,41)(28,40)(29,43)(31,44)(32,45)(36,46)(38,47)(42,48);(10,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,34,35)(24,29,38,42)(25,39,40,41)(30,44,45,46)(37,43,47,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(20,34)(25,40)(30,45)(33,35)(37,47)(39,41)(43,48)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6387582);