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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1371 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)(19,28)(20,34)(21,29)(22,40)(23,27)(24,30)(25,32)(26,38)(31,47)(33,45)(35,44)(36,43)(37,41)(39,46)(42,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,22,24,33,23,32)(18,25,27,45,30,40,29,28)(20,35,39,48,38,31,36,37)(26,42,46,44,34,41,43,47);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,24,38)(18,26,30,34)(19,31,33,35)(21,39,23,36)(22,37,32,48)(25,41,40,42)(27,46,29,43)(28,44,45,47);(2,6)(5,7)(11,16)(14,15)(17,21,24,23)(18,27,30,29)(19,32,33,22)(20,36,38,39)(25,28,40,45)(26,43,34,46)(31,37,35,48)(41,47,42,44);(2,6)(5,7)(17,22,21,33,24,32,23,19)(18,28,27,25,30,45,29,40)(20,37,39,35,38,48,36,31)(26,44,46,41,34,47,43,42)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=h,d^2=i,e^2=fg,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, 7514291);