Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,19)(14,16)(15,17)(18,20);(1,3,4,2)(6,10)(7,13)(11,18)(12,17)(14,20)(16,19);(7,13)(8,15)(11,19)(16,18);(5,7,9,13)(6,11,12,19)(8,14,15,20)(10,16,17,18)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,b^a=be,c^a=cf,h^a=hi,d^b=di,f^b=fhi,d^c=di,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260921, #72260953, #72260961, … |
| maximal normal subgroup of | #72260921, #72260953, #72260961, … |
| maximal quotient of | C2×#494357 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401553);