Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1306 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)(9,10,13,16)(11,14,12,15)(17,18,23,28)(19,29,30,43)(20,25,35,42)(21,27,22,26)(24,36,38,32)(31,45,33,48)(34,44,37,41)(39,46,40,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(19,30)(24,38)(29,43)(31,33)(32,36)(39,40)(45,48)(46,47);(9,11)(10,14)(12,13)(15,16)(17,19,22,32,23,30,21,36)(18,24,27,29,28,38,26,43)(20,33,34,46,35,31,37,47)(25,40,41,48,42,39,44,45);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20,23,35)(18,25,28,42)(19,31,30,33)(21,37,22,34)(24,39,38,40)(26,44,27,41)(29,45,43,48)(32,47,36,46);(10,16)(14,15)(17,21,23,22)(18,26,28,27)(19,32,30,36)(20,34,35,37)(24,29,38,43)(25,41,42,44)(31,46,33,47)(39,48,40,45)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6948978);