Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1384 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)(5,6,11,16)(7,17,21,35)(8,13,24,32)(9,25,18,33)(10,27,29,15)(12,22,31,20)(14,26,34,28)(19,30,23,36);(1,2)(3,4)(5,7,11,21)(6,12,16,31)(8,18,24,9)(10,22,29,20)(13,14,32,34)(15,35,27,17)(19,28,23,26)(25,36,33,30);(5,8,10,19,11,24,29,23)(6,13,15,36,16,32,27,30)(7,18,20,26,21,9,22,28)(12,14,17,25,31,34,35,33);(2,4)(5,9)(6,14)(7,19)(8,22)(10,28)(11,18)(12,30)(13,17)(15,25)(16,34)(20,24)(21,23)(26,29)(27,33)(31,36)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=egh,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297617);