Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499873 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1945 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1945 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(5,6)(7,14)(8,18)(9,20)(10,13)(11,16)(12,17)(15,19);(1,2)(3,4)(5,7,11,10)(6,12,16,18)(8,19,17,9)(13,20,14,15);(2,4)(6,13)(7,10)(8,17)(12,20)(14,16)(15,18);(5,8,11,17)(6,14,16,13)(7,9,10,19)(12,20,18,15)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72212257, #72266742, #72266849, … |
| maximal normal subgroup of | #72212257, #72266742, #72266849, … |
| maximal quotient of | #72212048, #72266746, #72266753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407072);