Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22)(23,28)(24,34)(25,30)(26,40)(27,31)(29,36)(32,43)(33,41)(35,46)(37,47)(38,44)(39,50)(42,49)(45,51)(48,52);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,27,26)(22,28,31,40)(24,32,37,36)(25,33,38,39)(29,34,43,47)(30,41,44,50)(35,45,48,49)(42,46,51,52);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,36,27,37,23,32)(22,29,28,34,31,43,40,47)(25,35,39,49,38,48,33,45)(30,42,41,46,44,51,50,52);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,25)(22,30)(23,33)(24,35)(26,39)(27,38)(28,41)(29,42)(31,44)(32,45)(34,46)(36,49)(37,48)(40,50)(43,51)(47,52);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(25,38)(30,44)(33,39)(35,48)(41,50)(42,51)(45,49)(46,52)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594852);