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 | #37680 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1650 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)(17,18,22,25)(19,28,30,23)(20,32,34,42)(21,26,36,27)(24,40,43,35)(29,44,33,47)(31,39,46,38)(37,45,41,48);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(17,19,22,30)(18,23,25,28)(20,33,34,29)(21,31,36,46)(24,41,43,37)(26,38,27,39)(32,44,42,47)(35,48,40,45);(2,6)(3,7)(10,13)(14,16)(17,20,21,35,22,34,36,40)(18,24,27,32,25,43,26,42)(19,29,31,45,30,33,46,48)(23,37,39,47,28,41,38,44);(9,11,12,15)(10,14,13,16)(18,25)(19,30)(20,34)(26,27)(31,46)(35,40)(37,41)(44,47);(3,7)(5,8)(17,21,22,36)(18,26,25,27)(19,31,30,46)(20,35,34,40)(23,38,28,39)(24,42,43,32)(29,45,33,48)(37,44,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgh,d^2=fi,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, 6671095);