Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511853 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32899 |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2153 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)(3,11)(4,15)(5,9)(6,10)(7,13)(8,17)(12,25)(14,24)(16,30)(18,22)(19,32)(20,28)(21,31)(23,27)(26,29);(1,3,6,14)(2,7,10,20)(4,12,17,27)(5,13,18,28)(8,19,15,29)(9,11,22,24)(16,26,31,32)(21,23,30,25);(5,18)(13,28)(16,31)(26,32);(1,4,6,17)(2,8)(3,12,14,27)(5,16,18,31)(7,19)(9,21)(10,15)(11,23)(13,26,28,32)(20,29)(22,30)(24,25)Polycyclic presentation:
2^9|b^2=i,d^2=h,b^a=be,c^a=cf,d^a=dg,h^a=hi,f^b=fhi,e^c=eh,g^d=gi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72233206, #72240644, #72240647, … |
| maximal normal subgroup of | #72233206, #72240644, #72240647, … |
| maximal quotient of | #72233100, #72233357, #72237946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419050);