Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(21,22)(23,32)(24,29)(25,30)(26,41)(27,31)(28,34)(33,49)(35,43)(36,52)(37,46)(38,47)(39,44)(40,48)(42,51)(45,50);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,34,27,35,25,39)(22,28,30,32,31,44,41,43)(24,36,38,50,37,33,40,51)(29,45,48,52,46,42,47,49);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,37)(22,29,31,46)(23,33,35,36)(25,40,26,38)(28,42,44,45)(30,48,41,47)(32,49,43,52)(34,51,39,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,37)(29,46)(33,36)(38,40)(42,45)(47,48)(49,52)(50,51);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,31,41)(23,34,35,39)(24,38,37,40)(28,43,44,32)(29,47,46,48)(33,50,36,51)(42,49,45,52)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437046);