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:
(6,13,16,30)(7,18,21,25)(8,23)(9,10)(11,24)(14,31)(15,28,26,17)(19,32)(20,35,27,22)(29,34);(1,2)(3,4)(5,6)(7,19,21,29)(8,15)(9,13)(10,26)(11,28)(12,16)(14,20,31,27)(17,24)(18,33,25,36)(22,32,35,34)(23,30);(2,4)(5,7,8,20,11,22,23,18,12,21,10,27,24,35,9,25)(6,14,13,29,17,33,26,32,16,31,30,19,28,36,15,34)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239851, #72239860, #72240855, … |
| maximal normal subgroup of | #72239851, #72239860, #72240855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56338);