Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | |
2540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6)(7,15)(8,20)(9,11)(10,24)(12,19)(13,22)(14,23)(16,26)(17,28)(18,29)(21,30)(25,27)(31,32)(33,36)(34,35);(1,2,4,3)(6,11)(7,16)(10,23)(12,15)(17,33)(18,19)(20,30)(22,25)(26,29)(28,31)(32,34)(35,36);(5,7)(6,12)(8,17)(9,18)(10,16)(11,26)(13,28)(14,29)(15,24)(19,23)(20,31)(21,34)(22,33)(25,32)(27,35)(30,36);(5,8,9,21)(6,13,11,27)(7,17,18,34)(10,25,23,22)(12,28,26,35)(14,30,24,20)(15,31,29,36)(16,32,19,33);(1,3,4,2)(6,11)(12,26)(13,27)(14,24)(15,29)(20,30)(28,35)(31,36)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425995);