Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418504 |
| 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 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17,20,29)(8,22,26,31)(9,15,10,14)(12,21,28,18)(13,24,32,25)(19,35,36,27)(23,30,34,33);(1,2,3,4)(5,7,9,18,11,20,10,21)(6,12,14,17,16,28,15,29)(8,23,24,36,26,34,25,19)(13,30,31,35,32,33,22,27);(7,18,20,21)(12,17,28,29)(19,34,36,23)(27,33,35,30);(2,4)(5,8)(6,13)(7,19)(9,25)(10,24)(11,26)(12,27)(14,22)(15,31)(16,32)(17,33)(18,34)(20,36)(21,23)(28,35)(29,30)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330127);