Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1082 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)(5,6,11,18,12,19,8,15)(7,20,21,34,25,36,24,32)(9,27,10,30,29,17,26,16)(13,28,14,22,33,23,35,31);(2,4)(5,7)(6,13,19,33)(8,21)(9,28)(10,31)(11,24)(12,25)(14,18,35,15)(16,34,30,32)(17,36,27,20)(22,26)(23,29);(5,7)(6,14)(8,21)(9,23)(10,22)(11,24)(12,25)(13,18)(15,33)(16,20)(17,34)(19,35)(26,31)(27,32)(28,29)(30,36);(5,8,12,11)(6,15,19,18)(7,21,25,24)(9,26,29,10)(13,14,33,35)(16,17,30,27)(20,34,36,32)(22,23,31,28)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,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 | #72264764, #72264817, #72264874, … |
| maximal normal subgroup of | #72264764, #72264817, #72264874, … |
| maximal quotient of | C2×#494339, #72309751, #72309781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401535);