Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 | #36493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(21,22)(23,32)(24,35)(25,30)(26,31)(27,29)(28,33)(34,36);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,24)(22,28,29,36)(25,33,26,34)(30,35,31,32);(6,10)(11,17)(12,18)(19,20)(21,24,27,23)(22,28,29,36)(25,34,26,33)(30,35,31,32);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,29)(28,36)(30,31)(32,35);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,29,31)(23,33,24,34)(28,35,36,32)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,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, 6451271);