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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1596 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)(11,13)(12,14)(15,16)(17,18,21,27,22,26,33,23)(19,24,30,37,29,38,43,34)(20,25,31,39,32,40,44,35)(28,36,41,47,42,46,48,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(21,33)(24,34)(26,27)(28,41)(30,43)(32,44)(35,39)(37,38)(42,48)(45,47);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,24,26,38)(20,28,32,42)(21,30,33,43)(23,34,27,37)(25,36,40,46)(31,41,44,48)(35,45,39,47);(3,8)(5,7)(11,16)(13,15)(17,20,21,31,22,32,33,44)(18,25,27,39,26,40,23,35)(19,28,30,41,29,42,43,48)(24,36,37,47,38,46,34,45);(10,14)(13,15)(18,26)(19,29)(20,32)(23,27)(30,43)(31,44)(36,46)(45,47)Polycyclic presentation:
2^9|a^2=fg,c^2=hi,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 6987807);