Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1442 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)(17,18,22,25)(19,27,30,40)(20,24,31,38)(21,33,35,26)(23,34,37,29)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,19,22,30)(18,23,25,37)(20,28,31,42)(21,34,35,29)(24,36,38,47)(26,40,33,27)(32,46,44,43)(39,48,45,41);(2,6)(5,7)(17,20,22,31)(18,24,25,38)(19,28,30,42)(21,32,35,44)(23,36,37,47)(26,39,33,45)(27,41,40,48)(29,43,34,46);(9,12,13,11)(10,15,16,14)(18,25)(20,31)(23,37)(26,33)(27,40)(28,42)(32,44)(43,46)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fhi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6759779);