Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23422 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1262 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)(5,6,10,18,12,16,11,17)(7,13,9,28,20,15,21,30)(8,22,27,32,25,33,26,14)(19,34,24,29,23,36,35,31);(5,7,10,21,12,20,11,9)(6,13,17,28,16,15,18,30)(8,23,26,35,25,19,27,24)(14,29,22,34,32,31,33,36);(5,8)(6,14)(7,19)(9,24)(10,27)(11,26)(12,25)(13,29)(15,31)(16,32)(17,33)(18,22)(20,23)(21,35)(28,36)(30,34);(2,4)(5,9,10,20,12,21,11,7)(6,15,18,30,16,13,17,28)(8,24,27,23,25,35,26,19)(14,31,22,34,32,29,33,36)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=ef,f^2=i,h^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,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 337656);