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 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,24)(20,30)(21,33)(22,36)(23,27)(25,28)(26,34)(29,43)(31,45)(32,38)(35,39)(37,41)(40,48)(42,47)(44,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,29,34,35)(25,38,41,30)(26,39,33,43)(31,42,46,48)(40,45,47,44);(9,11)(10,13)(12,16)(14,15)(17,20,23,32)(18,25,27,41)(19,28,22,37)(21,31,34,46)(24,38,36,30)(26,40,33,47)(29,42,35,48)(39,45,43,44);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,29)(20,31)(22,35)(23,34)(24,39)(25,40)(27,33)(28,42)(30,44)(32,46)(36,43)(37,48)(38,45)(41,47);(2,6)(5,7)(10,14)(13,15)(18,27)(21,34)(24,36)(25,41)(29,35)(30,38)(31,46)(42,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^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, 7208738);