Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37700 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1649 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,23)(20,32)(21,25)(22,27)(24,36)(26,34)(28,44)(29,38)(30,39)(31,43)(33,47)(35,42)(37,46)(40,45)(41,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,33)(21,31)(22,29)(24,40)(25,43)(26,39)(27,38)(28,35)(30,34)(32,47)(36,45)(37,41)(42,44)(46,48);(2,6)(3,7)(10,13)(14,16)(17,20,34,48,22,35,21,36)(18,24,25,42,27,41,26,32)(19,28,30,45,29,33,31,46)(23,37,43,47,38,40,39,44);(9,11,12,15)(10,14,13,16)(19,29)(20,35)(23,38)(24,41)(30,31)(32,42)(36,48)(39,43);(3,7)(5,8)(17,21,22,34)(18,25,27,26)(19,30,29,31)(20,36,35,48)(23,39,38,43)(24,42,41,32)(28,45,33,46)(37,44,40,47)Polycyclic presentation:
2^9|c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670737);