Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,17)(8,23)(9,14)(10,15)(11,16)(12,21)(13,25)(18,28)(19,20)(22,32)(24,29)(26,30)(27,31)(33,34)(35,36);(1,2)(3,4)(5,7,11,22)(6,12,16,19)(8,24,26,18)(9,20,10,21)(13,29,31,28)(14,17,15,32)(23,33,30,36)(25,35,27,34);(2,4)(5,8)(6,13)(7,18)(9,25)(10,27)(11,26)(12,28)(14,30)(15,23)(16,31)(17,33)(19,29)(20,34)(21,35)(22,24)(32,36);(2,4)(5,6)(7,19)(8,13)(9,15)(10,14)(11,16)(12,22)(17,20)(18,29)(21,32)(23,25)(24,28)(26,31)(27,30)(33,34)(35,36);(5,9,11,10)(6,14,16,15)(7,20,22,21)(8,25,26,27)(12,17,19,32)(13,30,31,23)(18,34,24,35)(28,33,29,36)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72251227, #72312619, #72312814, … |
| maximal normal subgroup of | #72251227, #72312619, #72312814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344678);