Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,9,15,11,16,10,17)(7,12,19,14,20,29,21,28)(8,22,25,31,24,32,26,13)(18,33,34,36,23,30,35,27);(5,7)(6,12)(8,23,24,18)(9,19)(10,21)(11,20)(13,27,31,36)(14,15)(16,29)(17,28)(22,33,32,30)(25,35,26,34);(2,4)(5,8)(6,13)(7,18)(9,26)(10,25)(11,24)(12,27)(14,30)(15,32)(16,31)(17,22)(19,35)(20,23)(21,34)(28,33)(29,36);(5,7)(6,14)(8,18)(9,21)(10,19)(11,20)(12,15)(13,30)(16,28)(17,29)(22,36)(23,24)(25,35)(26,34)(27,32)(31,33)Polycyclic presentation:
2^9|a^2=f,b^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326259);