Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1391 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)(19,31)(20,35)(21,29)(22,25)(23,27)(24,30)(26,39)(28,34)(32,48)(33,43)(36,46)(37,45)(38,42)(40,47)(41,44);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,22,24,34,21,33)(18,25,29,31,30,43,27,28)(20,36,37,38,39,32,40,41)(26,44,45,48,35,42,47,46);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,26)(19,32)(21,40)(22,41)(23,37)(24,39)(25,42)(27,47)(28,48)(29,45)(30,35)(31,46)(33,38)(34,36)(43,44);(2,6)(5,7)(10,14)(13,16)(17,21,24,23)(18,27,30,29)(19,33,34,22)(20,37,39,40)(25,28,43,31)(26,45,35,47)(32,41,36,38)(42,46,44,48);(2,6)(5,7)(17,22,23,34,24,33,21,19)(18,28,29,25,30,31,27,43)(20,38,37,32,39,41,40,36)(26,46,45,44,35,48,47,42)Polycyclic presentation:
2^9|b^2=f,c^2=h,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,e^b=ei,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507648);