Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2071 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,24)(9,27)(10,15)(11,17)(12,20)(14,29)(16,18)(19,30)(21,33)(22,35)(23,31)(25,34)(26,32)(28,36);(5,7,18,22,11,23,10,28)(6,12,15,24,17,33,16,32)(8,25,20,19,26,29,21,9)(13,34,36,27,31,14,35,30);(5,8,18,20,11,26,10,21)(6,13,15,36,17,31,16,35)(7,19,22,29,23,9,28,25)(12,30,24,34,33,27,32,14)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250075, #72250080, #72250085, … |
| maximal normal subgroup of | #72250075, #72250080, #72250085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56525);