Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2770 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)(24,36)(25,30)(26,39)(27,31)(28,35)(29,37)(33,42)(34,44)(38,45)(40,43)(41,46)(47,50)(48,51)(49,52);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,34)(22,28,31,43)(24,33,38,48)(25,35,26,40)(29,42,46,51)(30,44,39,32)(36,47,45,52)(37,49,41,50);(6,10)(11,17)(12,18)(19,20)(21,24,27,38)(22,29,31,46)(23,33,34,48)(25,37,26,41)(28,42,43,51)(30,45,39,36)(32,47,44,52)(35,49,40,50);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(23,34)(28,43)(32,44)(33,48)(35,40)(42,51)(47,52)(49,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,31,39)(23,35,34,40)(24,37,38,41)(28,44,43,32)(29,45,46,36)(33,49,48,50)(42,52,51,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451424);