Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,17,28,16)(7,18,22,32,23,29,36,33)(8,24,9,14,26,12,25,30)(13,20,34,27,31,19,35,21);(6,12)(7,19)(8,25)(9,26)(10,28)(13,31)(14,15)(16,30)(17,24)(18,32)(20,23)(21,36)(22,27)(29,33);(2,4)(5,7,8,20,10,22,9,27,11,23,26,19,28,36,25,21)(6,13,14,32,16,35,24,18,17,31,30,33,15,34,12,29)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=h,g^2=i,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,g^b=gi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239851, #72239860, #72239863, … |
| maximal normal subgroup of | #72239851, #72239860, #72239863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55736);