Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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)(21,22,25,31,26,30,38,44)(23,32,35,43,34,28,41,27)(24,29,39,48,40,47,49,52)(33,46,36,50,37,45,51,42);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23,26,34)(22,27,30,43)(24,36,40,51)(25,41,38,35)(28,44,32,31)(29,46,47,45)(33,49,37,39)(42,52,50,48);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,23)(22,28)(24,37)(25,35)(26,34)(27,31)(29,46)(30,32)(33,40)(36,49)(38,41)(39,51)(42,52)(43,44)(45,47)(48,50);(8,15)(11,13)(16,20)(18,19)(21,24,38,49,26,40,25,39)(22,29,44,52,30,47,31,48)(23,33,41,51,34,37,35,36)(27,42,28,45,43,50,32,46);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(22,30)(23,34)(29,47)(31,44)(33,37)(35,41)(36,51)(48,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=gh,d^2=fi,e^2=hi,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, 7105800);