Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,15)(8,18)(9,13)(10,14)(11,17)(12,16)(19,20);(1,2)(3,4)(5,7,10,16)(6,11,14,18)(8,9,17,19)(12,13,15,20);(2,4)(7,16)(8,17)(12,15)(13,20);(5,8,10,17)(6,12,14,15)(7,9,16,19)(11,13,18,20)Polycyclic presentation:
2^9|b^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,d^c=di,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72232683, #72233116, #72233436, … |
| maximal normal subgroup of | #72232683, #72233116, #72233436, … |
| maximal quotient of | #72232603, #72233111, #72233433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406996);