Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #149515 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1452 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | #346 | C23 |
| socle | C2 | #11415 |
| prime radical | C23 | #347 |
| LCS | G ▷ C2×C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,19,8,18,11,21)(3,12,13,23)(4,15)(10,25,26,31,22,32,24,20)(14,29,28,27)(16,30);(1,2,3,10,4,7,14,20,5,11,12,26,16,21,27,25,6,8,13,22,15,18,28,31,17,9,23,24,30,19,29,32);(2,8)(3,13)(7,18)(9,11)(12,23)(14,28)(19,21)(27,29)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bg,e^a=eh,g^a=gi,c^b=cf,d^b=dh,f^b=fi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214538, #72214568, #72214589, … |
| maximal normal subgroup of | #72214538, #72214568, #72214589, … |
| maximal quotient of | #72216250, #72217185, C2×#149515 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56711);