Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 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)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,26)(21,27)(22,39)(23,29)(24,30)(28,31)(32,41)(33,43)(34,44)(35,45)(36,46)(37,42)(38,48)(40,47);(1,3)(2,5)(4,8)(6,7)(17,19,24,33)(18,25,30,43)(20,34,37,38)(21,32,23,40)(22,36,31,35)(26,44,42,48)(27,41,29,47)(28,45,39,46);(10,13)(14,16)(17,20,24,37)(18,26,30,42)(19,31,33,22)(21,36,23,35)(25,28,43,39)(27,46,29,45)(32,38,40,34)(41,48,47,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,24,23)(18,27,30,29)(19,32,33,40)(20,35,37,36)(22,38,31,34)(25,41,43,47)(26,45,42,46)(28,44,39,48);(3,8)(5,7)(17,22,24,31)(18,28,30,39)(19,20,33,37)(21,38,23,34)(25,42,43,26)(27,44,29,48)(32,35,40,36)(41,46,47,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=gi,e^2=i,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,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6602927);