Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2303 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)(5,6,10,14,9,16,11,15)(7,12,20,26,19,27,23,17)(8,13,22,28,21,29,24,30)(18,25,33,31,32,36,34,35);(1,2)(3,4)(5,7,11,20,9,19,10,23)(6,12,15,26,16,27,14,17)(8,18,22,34,21,32,24,33)(13,25,28,35,29,36,30,31);(2,4)(5,6)(7,17)(8,13)(9,16)(10,15)(11,14)(12,23)(18,31)(19,26)(20,27)(21,29)(22,30)(24,28)(25,33)(32,35)(34,36);(5,8)(6,13)(7,18)(9,21)(10,24)(11,22)(12,25)(14,30)(15,28)(16,29)(17,31)(19,32)(20,34)(23,33)(26,35)(27,36)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280919, #72281024, #72281366, … |
| maximal normal subgroup of | #72280919, #72281024, #72281366, … |
| maximal quotient of | #72313570 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273828);