Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622911 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C25 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4030 conjugacy classes of subgroups; only the 568 normal subgroups are shown.
This group has 4030 conjugacy classes of subgroups — too many to draw. Only its 568 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,11)(4,15)(5,12)(6,10)(7,16)(8,13)(9,14);(1,3,8,7)(2,9,13,15)(4,5,14,6)(10,11,12,16);(1,4,8,14)(2,10,13,12)(3,6,7,5)(9,16,15,11);(1,3,8,7)(2,11,13,16)(4,6,14,5)(9,12,15,10);(1,5,8,6)(2,12,13,10)(3,14,7,4)(9,16,15,11)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,e^d=ei,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72209905, #72209907, #72234102, … |
| maximal normal subgroup of | #72209905, #72209907, #72234102, … |
| maximal quotient of | #72213335, #72227583, #72227830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530110);