Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1094 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)(7,19)(8,22)(9,25)(10,17)(11,18)(12,16)(13,21)(14,24)(15,29)(20,30)(23,31)(26,35)(27,34)(28,33)(32,36);(5,7,11,14)(6,13)(8,18)(9,20,28,31)(10,21,12,22)(15,30)(16,19)(17,24)(23,34)(25,26)(27,35,29,32)(33,36);(2,4)(5,8)(6,14)(7,18)(9,26)(10,19)(11,13)(12,24)(15,32)(16,21)(17,22)(20,25)(23,27)(28,36)(29,30)(31,33)(34,35);(5,9,11,28)(6,15,18,34)(7,20,14,31)(8,23,13,30)(10,29,12,27)(16,25,17,33)(19,26,24,36)(21,32,22,35)Polycyclic presentation:
2^9|b^2=egh,d^2=i,e^2=i,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,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378648);