Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,10,16,11,12)(7,13,20,25,19,28,17,27)(8,21,24,14,23,30,22,29)(18,32,33,36,34,35,31,26);(1,2)(3,4)(6,12)(7,17)(8,22)(9,11)(14,29)(15,16)(19,20)(23,24)(25,27)(26,32)(31,33)(35,36);(2,4)(5,7,10,19)(6,13,16,28)(8,18,23,34)(9,17,11,20)(12,25,15,27)(14,26,29,36)(21,32,30,35)(22,33,24,31);(5,8)(6,14)(7,18)(9,24)(10,23)(11,22)(12,21)(13,26)(15,30)(16,29)(17,31)(19,34)(20,33)(25,32)(27,35)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265644, #72265649, #72265683, … |
| maximal normal subgroup of | #72265644, #72265649, #72265683, … |
| maximal quotient of | #72238861 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290255);