Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1790 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2398 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)(3,15,19,28)(4,12,6,22)(5,11)(7,13)(8,14)(9,16)(10,20,27,21)(17,24,18,30)(23,26)(25,31)(29,32);(1,3)(2,9)(4,16)(5,17)(6,23)(7,18)(8,19)(10,24)(11,25)(12,28)(13,29)(14,26)(15,22)(20,31)(21,32)(27,30);(1,4)(2,10)(3,16)(5,20)(6,8)(7,21)(9,24)(11,12)(13,22)(14,27)(15,29)(17,31)(18,32)(19,23)(25,28)(26,30);(1,5,8,7)(2,11,14,13)(3,17,19,18)(4,20,6,21)(9,25,26,29)(10,12,27,22)(15,24,28,30)(16,31,23,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,f^c=fi,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236900, #72237540, #72237551, … |
| maximal normal subgroup of | #72236900, #72237540, #72237551, … |
| maximal quotient of | #72260435, #72261217, #72261225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400816);