Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140211 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| 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)(7,19,23,36)(8,25)(9,28)(10,15)(11,27)(12,18)(13,20,32,21)(14,17)(16,29)(22,35,31,33)(24,34,30,26);(1,2)(3,4)(5,7,11,30,12,23,14,24)(6,13,27,33,18,32,17,35)(8,26,28,36,15,34,16,19)(9,22,10,21,29,31,25,20);(2,4)(5,8,10,6,14,28,9,27,12,15,25,18,11,16,29,17)(7,13,31,26,30,35,20,19,23,32,22,34,24,33,21,36)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=h,f^2=hi,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=fhi,g^b=gi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250421, #72250458, #72250464, … |
| maximal normal subgroup of | #72250421, #72250458, #72250464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56372);