Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #28349 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2027 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,27,31)(23,33,35,29)(24,28,37,44)(25,30,39,48)(26,41,42,32)(34,49,38,45)(36,43,51,46)(40,52,50,47);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,24,27,35,42,37)(22,28,32,33,31,44,41,29)(25,38,40,51,39,34,50,36)(30,46,47,45,48,43,52,49);(6,11)(7,13)(12,15)(14,19)(21,24,26,35,27,37,42,23)(22,29,32,28,31,33,41,44)(25,36,40,38,39,51,50,34)(30,45,47,43,48,49,52,46);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,34)(24,36)(26,40)(27,39)(28,43)(29,45)(31,48)(32,47)(33,49)(35,38)(37,51)(41,52)(42,50)(44,46);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(25,39)(28,44)(29,33)(32,41)(34,38)(36,51)(40,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406118);