Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
954 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)(5,6,11,17,12,19,29,18)(7,14,23,35,25,36,24,32)(8,26,27,34,21,13,22,33)(9,28,30,15,20,16,10,31);(2,4)(5,7,11,23,12,25,29,24)(6,13,18,34,19,26,17,33)(8,20,27,10,21,9,22,30)(14,15,32,28,36,31,35,16);(5,8)(6,14)(7,20)(9,25)(10,23)(11,27)(12,21)(13,15)(16,33)(17,35)(18,32)(19,36)(22,29)(24,30)(26,31)(28,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239804, #72239807, #72239813, … |
| maximal normal subgroup of | #72239804, #72239807, #72239813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53999);