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 | |
4881 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)(5,6,12,16)(7,10,15,11)(8,19,22,17)(9,14,28,25)(13,24,20,23)(18,27,30,26)(21,32,34,31)(29,35,33,36);(1,2,3,4)(5,7,10,16,12,15,11,6)(8,20,23,17,22,13,24,19)(9,25,26,18,28,14,27,30)(21,31,35,33,34,32,36,29);(5,8,11,23,12,22,10,24)(6,13,15,19,16,20,7,17)(9,21,26,36,28,34,27,35)(14,29,30,32,25,33,18,31);(5,9)(6,14)(7,18)(8,21)(10,27)(11,26)(12,28)(13,29)(15,30)(16,25)(17,31)(19,32)(20,33)(22,34)(23,36)(24,35);(8,22)(13,20)(17,19)(21,34)(23,24)(29,33)(31,32)(35,36)Polycyclic presentation:
2^9|a^2=i,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350489);