Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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×#3483 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2683 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 2683 conjugacy classes of subgroups — too many to draw. Only its 314 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,16,11,17,7,15)(8,13,9,12,19,20,18,14);(6,12)(7,10)(9,18)(13,15)(14,17)(16,20);(2,4)(5,7)(8,18)(9,19)(10,11)(12,14)(15,16);(5,8,11,19)(6,13,17,20)(7,18,10,9)(12,15,14,16)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^e=fi,g^e=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261011, #72261027, #72261055, … |
| maximal normal subgroup of | #72261011, #72261027, #72261055, … |
| maximal quotient of | #72261886 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400807);