Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2322 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)(4,7)(5,17)(6,10)(8,13)(9,19)(12,25)(14,24)(15,21)(16,29)(18,23)(20,27)(22,28)(26,32)(30,31);(2,7)(8,20)(9,22)(10,21)(11,25)(17,29)(23,31)(24,32);(4,15)(12,26)(16,28)(27,30);(1,3,6,14)(2,8,10,23)(4,12,15,26)(5,18,19,13)(7,20,21,31)(9,24,17,11)(16,30,28,27)(22,32,29,25)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,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 | #72237525, #72237540, #72260522, … |
| maximal normal subgroup of | #72237525, #72237540, #72260522, … |
| maximal quotient of | #72261163, #72281274, #72281473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419016);