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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 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)(21,22,26,30,27,31,40,25)(23,32,34,43,35,45,41,28)(24,29,38,36,37,44,51,39)(33,46,49,47,48,50,52,42);(7,13)(8,16)(10,17)(11,15)(21,23,27,35)(22,28,31,43)(24,33,37,48)(25,32,30,45)(26,41,40,34)(29,42,44,47)(36,50,39,46)(38,52,51,49);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,37)(28,42)(30,36)(31,44)(32,46)(34,49)(35,48)(40,51)(41,52)(43,47)(45,50);(7,13)(10,17)(14,19)(18,20)(21,25,40,31,27,30,26,22)(23,28,41,45,35,43,34,32)(24,36,51,29,37,39,38,44)(33,47,52,46,48,42,49,50);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,37)(29,44)(33,48)(36,39)(38,51)(42,47)(46,50)(49,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,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=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103971);