Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2077 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,20,23,33)(8,13)(9,15)(10,16)(11,29)(12,19)(14,22,31,27)(17,25)(18,26)(21,32,28,36)(24,30,35,34);(1,2)(3,4)(6,13,19,16)(7,21,23,28)(8,25)(9,10)(11,26)(15,29,17,18)(20,34)(22,35,27,24)(30,33)(32,36);(2,4)(5,7,9,27,11,24,10,28,12,23,25,22,26,35,8,21)(6,14,13,20,18,32,17,34,19,31,16,33,29,36,15,30)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,h^b=hi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239863, #72240848, #72240855, … |
| maximal normal subgroup of | #72239863, #72240848, #72240855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56364);