Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1674 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)(5,6,12,8,11,15,10,22)(7,16,18,13,19,20,26,29)(9,14,24,34,25,31,27,21)(17,32,36,28,33,30,23,35);(5,7)(6,13)(8,20)(9,23)(10,26)(11,19)(12,18)(14,30)(15,29)(16,22)(17,24)(21,28)(25,36)(27,33)(31,32)(34,35);(2,4)(5,8,10,15,11,22,12,6)(7,16,18,29,19,20,26,13)(9,21,24,31,25,34,27,14)(17,30,23,35,33,32,36,28);(5,9,10,24,11,25,12,27)(6,14,8,21,15,31,22,34)(7,17,18,23,19,33,26,36)(13,28,16,30,29,35,20,32)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327689);