Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140211 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
654 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)(6,12,15,25)(7,17,21,26)(8,9)(10,27)(14,16,23,29)(18,31)(19,22,20,36)(24,28)(30,34)(32,35);(5,6)(7,18)(8,23)(9,25)(10,29)(11,15)(12,28)(13,20)(14,24)(16,27)(17,35)(19,33)(21,30)(22,34)(26,32)(31,36);(2,4)(5,7,8,19,10,22,9,26,11,21,24,20,27,36,28,17)(6,13,12,30,16,32,14,34,15,33,25,18,29,35,23,31)Polycyclic presentation:
2^9|a^2=g,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=fhi,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 | #72239807, #72239809, #72239826, … |
| maximal normal subgroup of | #72239807, #72239809, #72239826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55755);