Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | #36020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,45)(29,38)(31,48)(32,41)(33,42)(34,43)(35,39)(36,44)(37,46)(40,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30)(18,23,25,27)(20,31,32,47)(22,35,34,29)(24,40,41,48)(26,38,43,39)(28,36,46,33)(37,44,45,42);(10,14)(13,15)(17,20,21,32)(18,24,25,41)(19,28,30,46)(22,36,34,33)(23,37,27,45)(26,44,43,42)(29,47,35,31)(38,40,39,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(20,32)(23,27)(26,43)(28,46)(31,47)(33,36)(38,39);(2,5)(6,8)(11,16)(13,15)(18,25)(19,29)(22,34)(23,38)(24,41)(27,39)(28,47)(30,35)(31,46)(33,36)(37,40)(45,48)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606446);