Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,28)(19,29,33,24)(20,34,36,26)(21,27,23,39)(25,38,45,35)(30,46,31,41)(32,43,37,42)(40,48,44,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,29)(20,31,36,30)(21,37,23,32)(25,44,45,40)(26,41,34,46)(27,43,39,42)(35,47,38,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,45)(19,30,33,31)(21,35,23,38)(24,40,29,44)(26,39,34,27)(32,47,37,48)(41,43,46,42);(3,8)(5,7)(10,14)(11,16)(17,20)(18,26)(19,31)(21,38)(22,36)(23,35)(24,41)(25,27)(28,34)(29,46)(30,33)(32,47)(37,48)(39,45)(40,42)(43,44);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,27,28,39)(19,32,33,37)(20,35,36,38)(24,42,29,43)(25,26,45,34)(30,47,31,48)(40,41,44,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7208745);