Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,7,12,11,16,9,15)(8,14,17,13,10,19,18,20);(1,2,3,4)(6,12)(7,9)(13,14)(15,16)(17,18)(19,20);(5,7)(6,13)(8,17)(9,11)(10,18)(12,19)(14,15)(16,20);(5,8,11,10)(6,14,16,19)(7,17,9,18)(12,13,15,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72267983, #72280155, #72280209, … |
| maximal normal subgroup of | #72267983, #72280155, #72280209, … |
| maximal quotient of | #72267984 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401812);