Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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×#3020 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1864 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1864 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,21)(9,14)(10,26)(11,17)(12,19)(13,23)(15,24)(16,27)(20,33)(22,31)(25,29)(28,32)(30,36)(34,35);(1,2)(3,4)(5,7,11,8)(6,12,17,31)(9,22,24,19)(10,20,27,23)(13,32,33,29)(14,18,15,21)(16,34,26,30)(25,36,28,35);(2,4)(5,8,11,7)(6,13,17,33)(9,22,24,19)(10,20,27,23)(12,29,31,32)(14,30,15,34)(16,18,26,21)(25,35,28,36);(6,14)(9,24)(10,27)(12,18)(13,30)(15,17)(16,32)(19,22)(20,23)(21,31)(26,29)(33,34)Polycyclic presentation:
2^9|b^2=i,c^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,e^b=ei,f^b=fh,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72212272, #72266731, #72266866, … |
| maximal normal subgroup of | #72212272, #72266731, #72266866, … |
| maximal quotient of | #72212048, #72266733, #72266763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407054);