Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483956 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,13)(7,15)(8,20,21,36)(10,24,23,14)(11,25)(12,29,30,22)(16,31)(17,28)(18,35)(19,32)(26,33)(27,34);(5,7,8,17,10,18,22,34,9,19,21,25,23,16,29,33)(6,11,12,15,14,27,20,31,13,28,30,32,24,26,36,35);(1,2,3,4);(7,16,19,18)(8,21)(11,26,28,27)(12,30)(15,31,32,35)(17,34,25,33)(20,36)(22,29)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=gi,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,g^b=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224815, #72224823, #72224824, … |
| maximal normal subgroup of | #72224815, #72224823, #72224824, … |
| maximal quotient of | #72225234, #72225270, #72225272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403321);