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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1759 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)(21,22,25,30,26,31,37,27)(23,28,34,42,33,41,47,38)(24,29,35,43,36,44,48,39)(32,40,45,51,46,50,52,49);(7,13)(8,16)(10,17)(11,15)(22,27)(24,35)(25,37)(28,38)(30,31)(32,45)(34,47)(36,48)(39,43)(41,42)(46,52)(49,51);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23)(22,28)(24,32)(25,34)(26,33)(27,38)(29,40)(30,42)(31,41)(35,45)(36,46)(37,47)(39,49)(43,51)(44,50)(48,52);(7,13)(10,17)(14,19)(18,20)(21,24,25,35,26,36,37,48)(22,29,30,43,31,44,27,39)(23,32,34,45,33,46,47,52)(28,40,42,51,41,50,38,49);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,33)(28,41)(32,46)(34,47)(38,42)(40,50)(45,52)(49,51)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103953);