Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | |
1632 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)(23,33,41,46,35,29,36,43)(24,37,42,48,40,30,38,44)(34,45,50,51,39,47,49,52);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(22,28)(24,38)(25,27)(29,43)(31,32)(33,46)(34,49)(36,41)(39,50)(40,42)(44,48)(51,52);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,23,26,35)(22,29,32,33)(24,39,40,34)(25,36,27,41)(28,43,31,46)(30,47,37,45)(38,50,42,49)(44,52,48,51);(8,15)(11,13)(16,20)(18,19)(21,24,25,38,26,40,27,42)(22,30,31,48,32,37,28,44)(23,34,36,49,35,39,41,50)(29,45,46,52,33,47,43,51);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(23,35)(29,33)(34,39)(36,41)(43,46)(45,47)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=gh,d^2=f,e^2=h,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=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105813);