Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| 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 | #62759 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,19,26)(3,8,14,24,12,22,27,31)(4,9,17,25,15,7,18,21)(13,23,28,32,16,20,29,30);(2,7)(4,15)(8,20)(9,11)(10,25)(13,16)(17,18)(21,26)(22,23)(24,32)(28,29)(30,31);(1,3)(2,8)(4,16)(5,14)(6,12)(7,20)(9,23)(10,24)(11,22)(13,15)(17,29)(18,28)(19,27)(21,30)(25,32)(26,31);(3,12)(8,22)(13,16)(14,27)(20,23)(24,31)(28,29)(30,32)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,e^b=ei,f^b=fhi,g^b=ghi,d^c=di,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261408, #72261432, #72261563, … |
| maximal normal subgroup of | #72261408, #72261432, #72261563, … |
| maximal quotient of | #72303481, #72303482, #72303486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400830);