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 | C2×#3018 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1942 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1942 conjugacy classes of subgroups — too many to draw. Only its 150 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)(7,17)(8,15)(9,12)(10,20)(11,16)(13,19)(14,18);(1,2)(3,4)(6,12)(8,10)(9,18)(13,20)(14,16)(15,17);(2,4)(5,7,11,19)(6,12,16,14)(8,9,10,18)(13,20,17,15);(5,8,11,10)(6,13,16,17)(7,18,19,9)(12,15,14,20)Polycyclic presentation:
2^9|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 | #72227997, #72228003, #72228005, … |
| maximal normal subgroup of | #72227997, #72228003, #72228005, … |
| maximal quotient of | #72262029, #72268020, #72324327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407073);