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 | |
1393 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)(17,18,23,28)(19,29,31,26)(20,25,32,41)(21,24,36,40)(22,37,38,27)(30,42,45,44)(33,47,35,39)(34,48,46,43);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,21,23,31,38,36)(18,24,27,29,28,40,37,26)(20,30,34,35,32,45,46,33)(25,39,43,42,41,47,48,44);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20,23,32)(18,25,28,41)(19,30,31,45)(21,35,36,33)(22,34,38,46)(24,39,40,47)(26,44,29,42)(27,43,37,48);(2,6)(5,7)(10,14)(13,16)(19,31)(20,32)(21,36)(24,40)(25,41)(26,29)(34,46)(43,48);(2,6)(5,7)(17,21,22,31,23,36,38,19)(18,26,27,24,28,29,37,40)(20,33,34,30,32,35,46,45)(25,42,43,47,41,44,48,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,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^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507693);