Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,15)(8,19)(9,22)(10,14)(11,16)(12,20)(13,23)(17,26)(18,29)(21,30)(24,27)(25,28)(31,33)(32,35)(34,36);(1,2,4,3)(7,16)(9,23)(11,15)(12,28)(13,22)(17,33)(18,24)(20,25)(26,31)(27,29)(32,34)(35,36);(5,7)(6,11)(8,17)(9,24)(10,18)(12,26)(13,29)(14,27)(15,22)(16,23)(19,31)(20,32)(21,34)(25,33)(28,36)(30,35);(5,8,10,21)(6,12,14,28)(7,17,18,34)(9,25,23,20)(11,26,27,36)(13,30,22,19)(15,31,29,35)(16,32,24,33);(1,3,4,2)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425979);