Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,7,16,14)(8,12,21,26,20,17,29,25)(9,13,23,28,22,18,30,27)(19,24,32,35,33,31,36,34);(5,7)(6,11)(8,17)(9,18)(10,14)(12,20)(13,22)(15,16)(19,31)(21,25)(23,27)(24,33)(26,29)(28,30)(32,34)(35,36);(2,4)(5,8)(6,12)(7,17)(9,19)(10,21)(11,20)(13,24)(14,25)(15,26)(16,29)(18,31)(22,33)(23,32)(27,34)(28,35)(30,36);(5,9,10,23,11,22,16,30)(6,13,15,28,7,18,14,27)(8,19,21,32,20,33,29,36)(12,24,26,35,17,31,25,34);(8,20)(9,22)(12,17)(13,18)(21,29)(23,30)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,e^c=ei,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72320787 |
| maximal normal subgroup of | #72320787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279890);