Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| 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×D4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,13)(9,22)(10,15)(11,16)(12,19)(14,23)(18,27)(20,30)(21,28)(24,35)(25,26)(29,34)(31,33)(32,36);(1,2)(3,4)(5,7,11,8)(6,12,16,28)(9,20,24,18)(10,19,25,21)(13,15,17,26)(14,30,32,27)(22,29,35,31)(23,34,36,33);(2,4)(5,8,11,7)(6,13,16,17)(9,18,24,20)(10,21,25,19)(12,26,28,15)(14,31,32,29)(22,30,35,27)(23,33,36,34);(5,9,11,24)(6,14,16,32)(7,18,8,20)(10,23,25,36)(12,27,28,30)(13,29,17,31)(15,22,26,35)(19,33,21,34)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fh,d^c=di,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312239, #72312245, #72312272, … |
| maximal normal subgroup of | #72312239, #72312245, #72312272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407014);