Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,31,26,30,29,46)(23,32,36,43,35,34,41,27)(24,28,40,45,39,38,47,52)(33,48,49,44,37,50,51,42);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,35)(22,27,30,43)(24,37,39,33)(25,41,29,36)(28,44,38,42)(31,34,46,32)(40,49,47,51)(45,48,52,50);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,26,39)(22,28,30,38)(23,33,35,37)(25,40,29,47)(27,42,43,44)(31,45,46,52)(32,48,34,50)(36,49,41,51);(8,16)(11,20)(13,15)(17,19)(21,22,29,46,26,30,25,31)(23,34,41,43,35,32,36,27)(24,38,47,45,39,28,40,52)(33,48,51,42,37,50,49,44);(6,12)(7,14)(11,20)(13,15)(22,30)(23,35)(24,39)(31,46)(36,41)(40,47)(42,44)(48,50)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011993);