Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,19,23,36)(8,21,26,32)(9,15)(10,28)(11,31)(12,33)(13,18)(14,25,20,35)(16,30)(17,27)(22,34,29,24);(2,4)(5,7,12,20,13,23,9,14)(6,8,15,34,18,26,33,24)(10,29,30,21,27,22,11,32)(16,35,28,19,31,25,17,36);(5,8)(6,14)(7,15)(9,24)(10,25)(11,19)(12,34)(13,26)(16,21)(17,29)(18,20)(22,28)(23,33)(27,35)(30,36)(31,32);(5,9,13,12)(6,15,18,33)(7,20,23,14)(8,24,26,34)(10,11,27,30)(16,28,31,17)(19,35,36,25)(21,22,32,29)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285780, #72319775, #72319823, … |
| maximal normal subgroup of | #72285780, #72319775, #72319823, … |
| maximal quotient of | #72285792, #72285913, #72287008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401532);