Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493621 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1790 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #62874 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,12,4,10)(3,13,16,31,17,28,14,30)(5,11,18,27,21,29,20,26)(8,22,24,15,25,32,23,19);(1,3)(2,8)(4,14)(5,19)(6,16)(7,17)(9,23)(10,24)(11,28)(12,25)(13,29)(15,21)(18,22)(20,32)(26,30)(27,31);(2,9)(4,6)(8,23)(10,12)(11,27)(13,30)(14,16)(18,20)(22,32)(24,25)(26,29)(28,31);(1,4,7,6)(2,10,12,9)(3,14,17,16)(5,18,21,20)(8,24,25,23)(11,27,29,26)(13,30,28,31)(15,32,19,22)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fi,g^c=gi,f^e=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72234207, #72234212, #72261065, … |
| maximal normal subgroup of | #72234207, #72234212, #72261065, … |
| maximal quotient of | #72238625, #72286501 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400823);