Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| 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 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2061 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)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19);(1,2)(3,4)(5,7)(6,11)(8,10)(9,17)(12,19)(13,15)(14,20)(16,18);(2,4)(5,8,10,7)(6,12,14,15)(9,16,18,17)(11,19,20,13);(5,9)(7,16)(8,17)(10,18)(11,20)(12,15)Polycyclic presentation:
2^9|c^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,d^b=di,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 | #72233167, #72234719, #72234763, … |
| maximal normal subgroup of | #72233167, #72234719, #72234763, … |
| maximal quotient of | #72233163, #72233446, #72233482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407009);