Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2366 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6,8,12)(3,9)(7,14,15,10)(11,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(1,2,4,12,5,8,13,6)(3,10,11,7,9,14,16,15)(19,27)(23,31)(26,32)(28,29);(17,19,21,29,22,27,20,28)(18,23,24,26,25,31,30,32);(1,3,4,11,5,9,13,16)(2,7,6,10,8,15,12,14)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287574, #72287601, #72287611, … |
| maximal normal subgroup of | #72287574, #72287601, #72287611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138741);