Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493562 |
| 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×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62874 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,17,25)(3,8,13,23,11,22,26,31)(4,7,16,21,15,20,18,24)(12,19,27,30,14,29,28,32);(2,7)(5,17)(8,19)(9,21)(10,20)(13,26)(16,18)(22,29)(23,30)(24,25)(27,28)(31,32);(1,3)(2,8)(4,14)(5,13)(6,11)(7,19)(9,23)(10,22)(12,15)(16,28)(17,26)(18,27)(20,29)(21,30)(24,32)(25,31);(3,11)(8,22)(12,14)(13,26)(19,29)(23,31)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,d^c=di,e^c=eh,f^e=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72287178, #72303744, #72303750, … |
| maximal normal subgroup of | #72287178, #72303744, #72303750, … |
| maximal quotient of | #72287206, #72340322 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400766);