Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498581 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23188 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33717 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,14,25,26,8,15,30,12,23,13,4,22)(3,16,18,27,20,32,11,9,21,5,17,29,19,28,31,24);(1,3,8,21)(2,9)(4,17,25,18)(5,26)(6,11,30,31)(7,19,23,20)(10,28)(12,32)(13,29)(14,27)(15,24)(16,22);(1,4,23,30,8,25,7,6)(2,10,14,26,15,12,13,22)(3,17,20,31,21,18,19,11)(5,24,32,29,16,9,28,27);(1,5,6,27,7,28,25,9,8,16,30,29,23,32,4,24)(2,11,12,21,14,17,22,19,15,31,10,3,13,18,26,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214308, #72214332, … |
| maximal normal subgroup of | #72214039, #72214308, #72214332, … |
| maximal quotient of | #72216421, #72227102, #72227103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405778);