Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #23810 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,28,40,27,25,26,31)(23,32,41,29,35,45,34,38)(24,30,42,52,36,39,37,44)(33,46,50,43,48,47,49,51);(7,13)(8,16)(10,17)(11,15)(21,23,27,35)(22,29,25,38)(24,33,36,48)(26,41,28,34)(30,43,39,51)(31,45,40,32)(37,50,42,49)(44,47,52,46);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24,27,36)(22,30,25,39)(23,33,35,48)(26,37,28,42)(29,43,38,51)(31,44,40,52)(32,46,45,47)(34,49,41,50);(7,13)(10,17)(14,19)(18,20)(21,25,28,31,27,22,26,40)(23,32,41,29,35,45,34,38)(24,30,42,52,36,39,37,44)(33,47,50,51,48,46,49,43);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,36)(30,39)(33,48)(37,42)(43,51)(44,52)(46,47)(49,50)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=ghi,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=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103975);