Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,22)(9,13)(10,15)(11,19)(12,25)(14,27)(17,30)(18,21)(20,29)(23,33)(24,32)(26,31)(28,35)(34,36);(2,4)(5,7)(6,11)(8,23)(9,19)(10,20)(12,30)(13,16)(14,29)(15,18)(17,26)(21,27)(22,28)(24,35)(25,34)(31,36)(32,33);(5,8,10,25,9,24,27,26)(6,12,14,22,13,31,15,32)(7,17,20,23,19,34,21,35)(11,28,29,36,16,33,18,30);(5,6)(7,18)(8,12)(9,13)(10,14)(11,20)(15,27)(16,21)(17,30)(19,29)(22,25)(23,28)(24,31)(26,32)(33,35)(34,36);(8,24)(12,31)(17,34)(22,32)(23,35)(25,26)(28,33)(30,36)Polycyclic presentation:
2^9|a^2=fi,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350154);