Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1312 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,27,34,23)(19,28,31,35,30,24,37,39)(20,25,32,40,33,41,45,36)(29,42,43,46,44,38,47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(21,34)(24,35)(26,27)(28,39)(29,43)(31,37)(33,45)(36,40)(44,47)(46,48);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,24,27,28)(20,29,33,44)(21,31,34,37)(23,35,26,39)(25,38,41,42)(32,43,45,47)(36,46,40,48);(11,16)(13,15)(17,20,21,32,22,33,34,45)(18,25,26,40,27,41,23,36)(19,29,31,43,30,44,37,47)(24,38,39,48,28,42,35,46);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(19,30)(24,28)(29,44)(31,37)(35,39)(38,42)(43,47)(46,48)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=gh,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7000996);