Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439035 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,12)(7,17)(9,10)(13,26)(14,22)(15,16)(19,31)(20,21)(23,24)(27,35)(29,30)(33,34);(1,2,3,4)(5,6,9,12,11,16,10,15)(7,18,21,13,20,28,17,26)(8,22,23,14,25,29,24,30)(19,27,31,32,33,35,34,36);(5,7,9,17,11,20,10,21)(6,13,12,18,16,26,15,28)(8,19,24,31,25,33,23,34)(14,27,22,32,30,35,29,36);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,22)(13,27)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(26,35)(28,36);(7,20)(13,26)(17,21)(18,28)(19,33)(27,35)(31,34)(32,36)Polycyclic presentation:
2^9|b^2=f,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255901, #72256284, #72256289, … |
| maximal normal subgroup of | #72255901, #72256284, #72256289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350488);