Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(19,26)(20,25)(21,36)(22,37)(23,28)(24,32)(27,29)(30,40)(31,42)(33,39)(34,46)(35,47)(38,44)(41,43)(45,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,29,21,23,31,22,32)(18,24,37,42,28,36,27,26)(20,30,43,34,33,45,35,44)(25,38,47,48,39,46,41,40);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,33)(18,25,28,39)(19,30,31,45)(21,34,32,44)(22,35,29,43)(24,38,36,46)(26,40,42,48)(27,41,37,47);(2,6)(5,7)(11,16)(14,15)(19,31)(20,33)(21,32)(24,36)(25,39)(26,42)(35,43)(41,47);(2,6)(5,7)(17,21,29,31,23,32,22,19)(18,26,37,24,28,42,27,36)(20,34,43,45,33,44,35,30)(25,40,47,38,39,48,41,46)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=h,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,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514290);