Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62819 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2418 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)(3,11,13,28)(4,16)(5,10)(6,8)(7,12,23,29)(9,20)(14,27,15,26)(17,22,30,24)(18,32)(19,31)(21,25);(1,3)(2,7)(4,17)(5,15)(6,14)(8,22)(9,26)(10,24)(11,18)(12,19)(13,21)(16,28)(20,29)(23,25)(27,31)(30,32);(1,2,5,10)(3,7,15,24)(4,18,19,9)(6,8,21,25)(11,12,26,17)(13,23,14,22)(16,20,31,32)(27,30,28,29);(2,8)(7,22)(9,16)(10,25)(11,27)(18,31)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#186831, #72239238, #72260647, … |
| maximal normal subgroup of | C2×#186831, #72239238, #72260647, … |
| maximal quotient of | C2×#186831, #72238884, #72238887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94027);