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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22,27,28,26,32,25,31)(23,29,38,40,34,43,35,44)(24,30,39,41,37,45,36,46)(33,42,49,50,48,52,47,51);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(22,28)(24,36)(25,27)(29,40)(31,32)(33,47)(35,38)(37,39)(41,46)(43,44)(48,49)(50,51);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,23)(22,29)(24,33)(25,35)(26,34)(27,38)(28,40)(30,42)(31,44)(32,43)(36,47)(37,48)(39,49)(41,50)(45,52)(46,51);(8,15)(11,13)(16,20)(18,19)(21,24,27,39,26,37,25,36)(22,30,28,41,32,45,31,46)(23,33,38,49,34,48,35,47)(29,42,40,50,43,52,44,51);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(23,34)(24,37)(29,43)(30,45)(35,38)(36,39)(40,44)(41,46)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=gh,d^2=fi,e^2=h,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105824);