Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 | |
1351 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18)(19,31)(20,35)(21,29)(22,25)(23,27)(24,30)(26,39)(28,34)(32,45)(33,42)(36,47)(37,44)(38,43)(40,46)(41,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,33,24,34,23,22)(18,25,27,28,30,42,29,31)(20,36,40,38,39,32,37,48)(26,43,46,47,35,41,44,45);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,24,39)(18,26,30,35)(19,32,34,36)(21,40,23,37)(22,38,33,48)(25,41,42,43)(27,46,29,44)(28,45,31,47);(2,6)(5,7)(11,16)(14,15)(17,21,24,23)(18,27,30,29)(19,33,34,22)(20,37,39,40)(25,28,42,31)(26,44,35,46)(32,38,36,48)(41,47,43,45);(2,6)(5,7)(17,22,23,34,24,33,21,19)(18,28,29,25,30,31,27,42)(20,38,37,36,39,48,40,32)(26,45,44,41,35,47,46,43)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=hi,d^2=i,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514120);