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 | #60644 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1931 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)(7,17)(8,22)(9,16)(10,13)(11,15)(12,18)(14,26)(19,35)(20,29)(21,27)(23,31)(24,30)(25,32)(28,36)(33,34);(1,2,3,4)(5,7,9,18,10,21,11,20)(6,12,15,17,13,29,16,27)(8,23,24,36,25,34,26,19)(14,31,32,35,30,33,22,28);(6,13)(7,18,21,20)(12,27,29,17)(14,30)(15,16)(19,34,36,23)(22,32)(28,31,35,33);(2,4)(5,8)(6,14)(7,19)(9,26)(10,25)(11,24)(12,28)(13,30)(15,22)(16,32)(17,33)(18,34)(20,23)(21,36)(27,31)(29,35)Polycyclic presentation:
2^9|b^2=e,c^2=fi,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, 330126);