Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #23863 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(17,18,23,24,22,27,19,28)(20,31,29,25,33,42,34,40)(21,35,36,41,37,26,30,38)(32,45,43,48,47,44,46,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,36,37,30)(25,31,40,42)(26,41,35,38)(32,46,47,43)(39,45,48,44);(10,13)(11,15)(17,20,22,33)(18,25,27,40)(19,29,23,34)(21,32,37,47)(24,31,28,42)(26,39,35,48)(30,43,36,46)(38,44,41,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,32)(22,37)(23,36)(24,38)(25,39)(27,35)(28,41)(29,43)(31,44)(33,47)(34,46)(40,48)(42,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,27)(21,37)(24,28)(25,40)(30,36)(31,42)(32,47)(43,46)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7190893);