Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,22,25,29)(23,27,30,35)(24,26,32,28)(31,33,36,34);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(22,26)(23,30)(24,32)(27,34)(28,29)(33,35);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(21,23,25,30)(22,27,29,35)(24,31,32,36)(26,33,28,34);(7,17)(11,14)(15,16)(19,20)(22,28)(24,32)(26,29)(27,34)(31,36)(33,35)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591556);