Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,12)(8,20,22,31)(9,14)(10,25)(11,15)(13,23,30,24)(16,29,32,36)(17,27)(18,28)(19,34)(21,26,33,35);(5,7)(6,12)(8,21,22,33)(9,18)(10,19)(11,17)(13,29,30,36)(14,28)(15,27)(16,24,32,23)(20,35,31,26)(25,34);(5,8)(6,13)(7,16)(9,24)(10,23)(11,22)(12,26)(14,31)(15,30)(17,32)(18,21)(19,33)(20,25)(27,35)(28,36)(29,34);(1,3,4,2)(7,17)(12,27)(16,32)(18,19)(21,33)(26,35)(28,34)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224748, #72224755, #72224756, … |
| maximal normal subgroup of | #72224748, #72224755, #72224756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266398);