Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1226 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)(7,13)(8,11)(9,12)(10,15)(14,17)(16,20)(18,19);(1,3,4,2)(6,11)(8,17)(12,19)(13,20)(14,15)(16,18);(6,12)(7,9)(8,17)(11,19)(13,15)(14,20);(5,7,10,9)(6,13,15,12)(8,16,17,18)(11,20,14,19)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260924, #72260944, #72260953, … |
| maximal normal subgroup of | #72260924, #72260944, #72260953, … |
| maximal quotient of | #72250630, #72282612 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401588);