Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
502 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)(7,18,23,30)(8,15)(9,13)(10,29)(11,17)(12,20,33,27)(14,25)(16,28)(19,31,22,34)(21,36,24,32)(26,35);(1,2)(3,4)(5,7)(6,12,17,33)(8,24)(9,27)(10,19)(11,23)(13,34,14,31)(15,18,35,30)(16,36,29,32)(20,25)(21,26)(22,28);(2,4)(6,13,17,14)(7,19)(8,25)(9,26)(10,28)(12,30,33,18)(15,29,35,16)(21,24)(22,23)(31,36,34,32)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250347, #72250373, #72250382, … |
| maximal normal subgroup of | #72250347, #72250373, #72250382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56534);