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 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1903 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1903 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)(7,15)(8,16)(9,14)(10,19)(11,20)(12,17)(13,18);(1,2)(3,4)(5,7,12,11)(6,13,17,14)(8,18,10,9)(15,16,20,19);(2,4)(5,8,12,10)(6,14,17,13)(7,9,11,18)(15,19,20,16);(5,9,12,18)(6,15,17,20)(7,10,11,8)(13,19,14,16)Polycyclic presentation:
2^9|b^2=i,c^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, #72212272, #72213168, … |
| maximal normal subgroup of | #72212257, #72212272, #72213168, … |
| maximal quotient of | #72212208, #72213149, C2×#499878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407074);