Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2208 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 2208 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,12)(7,11)(9,16)(13,19)(14,18)(15,20);(5,6)(7,15)(8,13)(9,18)(10,14)(11,19)(12,16)(17,20);(2,4)(5,7)(8,16)(9,17)(10,11)(12,18)(15,19);(5,8,10,17)(6,13,14,20)(7,16,11,9)(12,19,18,15)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=eh,g^c=gi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250552, #72257236, #72257242, … |
| maximal normal subgroup of | #72250552, #72257236, #72257242, … |
| maximal quotient of | #72264899 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406099);