Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | #39131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,42)(29,38)(30,39)(31,46)(33,41)(35,40)(37,43)(44,47)(45,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,27,38)(20,28,33,44)(21,30,22,31)(25,37,40,48)(26,39,36,46)(32,42,41,47)(34,43,35,45);(11,16)(13,15)(17,20,21,35,23,33,22,34)(18,25,36,41,27,40,26,32)(19,28,30,45,29,44,31,43)(24,37,46,47,38,48,39,42);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(19,29)(20,33)(24,38)(25,40)(30,31)(32,41)(34,35)(39,46);(1,4,5,3)(2,6,7,8)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,34,33,35)(24,39,38,46)(25,41,40,32)(28,43,44,45)(37,47,48,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6790146);