Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417601 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2373 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)(7,17)(8,22)(9,10)(12,25)(13,28)(14,15)(18,27)(20,21)(23,24)(26,36)(29,30)(31,33);(1,2)(3,4)(5,6)(7,18)(8,13)(9,15)(10,14)(11,16)(12,20)(17,27)(19,32)(21,25)(22,28)(23,30)(24,29)(26,33)(31,36)(34,35);(5,7,11,21)(6,12,16,27)(8,19,23,34)(9,20,10,17)(13,26,30,36)(14,18,15,25)(22,31,24,33)(28,32,29,35);(2,4)(5,8)(6,13)(7,19)(9,24)(10,22)(11,23)(12,26)(14,28)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(25,35)(27,36)Polycyclic presentation:
2^9|c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 324808);