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 | #23853 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1609 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,28,21,27)(19,29,35,40,31,25,32,37)(20,26,36,38,34,41,33,42)(30,43,46,48,45,39,44,47);(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(21,23)(25,37)(27,28)(29,40)(30,44)(32,35)(34,36)(38,42)(45,46)(47,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,31)(18,25,28,29)(20,30,34,45)(21,32,23,35)(24,37,27,40)(26,39,41,43)(33,44,36,46)(38,47,42,48);(11,16)(13,15)(17,20,23,36,22,34,21,33)(18,26,24,38,28,41,27,42)(19,30,35,46,31,45,32,44)(25,39,37,47,29,43,40,48);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,31)(20,34)(25,29)(26,41)(32,35)(33,36)(37,40)(38,42)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 6997966);