Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1790 |
| prime radical | #62729 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,21,28)(3,11,14,29,12,25,19,31)(4,15,18,27,17,8,20,26)(7,22,24,16,23,30,32,13);(1,3)(2,7)(4,16)(5,19)(6,12)(8,25)(9,24)(10,23)(11,15)(13,17)(14,21)(18,22)(20,30)(26,31)(27,29)(28,32);(8,15)(11,25)(26,27)(29,31);(3,12)(7,23)(11,25)(13,16)(14,19)(22,30)(24,32)(29,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233438, #72234211, #72234212, … |
| maximal normal subgroup of | #72233438, #72234211, #72234212, … |
| maximal quotient of | #72238627, #72251986, #72284680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400786);