Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| 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 | |
702 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)(5,6,11,16)(7,18)(8,13,9,25)(10,27,23,15)(12,28,24,17)(14,26)(19,33)(20,32)(21,29)(22,31)(30,35)(34,36);(6,13)(7,19)(8,23)(9,10)(12,24)(14,30)(15,17)(16,25)(18,31)(20,36)(21,26)(22,35)(27,28)(32,33);(2,4)(5,7,8,20,24,36,10,19,11,22,9,26,12,21,23,35)(6,14,15,31,28,33,25,34,16,32,27,18,17,30,13,29)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=hi,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=dh,e^b=eh,f^b=fhi,h^b=hi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239851, #72239855, #72239863, … |
| maximal normal subgroup of | #72239851, #72239855, #72239863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55715);