Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469983 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,31)(23,28,32,36)(24,30,25,29)(27,34,35,33);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,32)(22,27,31,35)(24,33,25,34)(28,30,36,29);(1,2,3,4)(7,13)(10,17)(14,19)(18,20)(21,23)(22,28)(24,34)(25,33)(26,32)(27,29)(30,35)(31,36);(7,13)(10,17)(14,19)(18,20)(23,32)(27,35)(28,36)(33,34);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,26,25)(22,29,31,30)(23,33,32,34)(27,28,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404746);